# KPP fronts in shear flows with cut-off reaction rates

D. J. Needham and A. Tzella Address for correspondence: Prof. D. J. Needham
and Dr A. Tzella, School of Mathematics, University of Birmingham; email:
<EMAIL_ADDRESS>School of Mathematics, University of Birmingham,
Birmingham, B15 2TT, UK.

###### Abstract

We consider the effect of a shear flow which has, without loss of generality,
a zero mean flow rate, on a Kolmogorov–Petrovskii–Piscounov (KPP) type model
in the presence of a discontinuous cut-off at concentration $u=u_{c}$. In the
long-time limit, a permanent form travelling wave solution is established
which, for fixed $u_{c}>0$, is unique. Its structure and speed of propagation
depends on $A$ (the strength of the flow relative to the propagation speed in
the absence of advection) and $B$ (the square of the front thickness relative
to the channel width). We use matched asymptotic expansions to approximate the
propagation speed in the three natural cases $A\to\infty$, $A\to 0$ and
$A=O(1)$, with particular associated orderings on $B$, whilst $u_{c}\in(0,1)$
remains fixed. In all the cases that we consider, the shear flow enhances the
speed of propagation in a manner that is similar to the case without cut-off
($u_{c}=0$). We illustrate the theory by evaluating expressions (either
directly or through numerical integration) for the particular cases of the
plane Couette and Poiseuille flows.

Keywords: reaction-diffusion, permanent form travelling waves, cut-off
nonlinearity, shear flow, asymptotic expansions.

##  1 Introduction

We investigate travelling fronts for cut-off reaction–diffusion equations
evolving in an infinite channel in the presence of a shear flow. The model
equation that we consider describes the spatio-temporal evolution of the
scalar function $u$ denoting the concentration field of a dissolved species,
and takes the non-dimensional generic form

$u_{t}+A\,\alpha(y)u_{x}=u_{xx}+B\,u_{yy}+f_{c}(u),~{}~{}(x,y)\in\mathbb{R}\times(0,1),~{}t\in\mathbb{R^{+}}.$
(IBVPa) Here, the dimensionless spatial coordinate $y$ has been obtained by
scaling with the channel width $a$ whilst the dimensionless spatial coordinate
$x$ has been obtained by scaling with the diffusive length scale
$\sqrt{\kappa\tau}$ where $\kappa$ is the molecular diffusivity and $\tau$ is
the reaction timescale. The latter has also been used to introduce the
dimensionless time $t$. The cut-off reaction function
$f_{c}:\mathbb{R}\to\mathbb{R}$ is given by
$f_{c}(u)=\begin{cases}f(u),&\text{$u>u_{c}$}\\\ 0,&\text{$u\leq
u_{c}$}\end{cases}$ (IBVPb) where $f(u)$ is a normalised KPP-type reaction
function, named after the classical work by Fisher [11] and Kolmogorov,
Petrovskii, and Piskunov [15], which satisfies $\begin{split}f(0)=f(1)=0,\quad
f^{\prime}(0)=1,\quad&f^{\prime}(1)<0\quad\text{and}\quad f\in
C^{1}(\mathbb{R})\\\ 0<f(u)\leq u\quad\forall
u\in(0,1),\quad\quad&f(u)<0\quad\forall u\in(1,\infty).\end{split}$ (IBVPc)
Thus the reaction is effectively deactivated at points where the concentration
$u$ lies at or below a threshold value $u_{c}\in(0,1)$. For convenience
throughout, we introduce the shorthand notation $f_{c}\coloneqq
f_{c}(u_{c}^{+})$, the limiting value from above of $f_{c}(u)$ at $u=u_{c}$. A
prototypical example of such a KPP reaction function is the Fisher reaction
function [11] given by $f(u)=u(1-u)$. The flow function
$\alpha:[0,1]\to\mathbb{R}$ satisfies $\alpha\in C^{1}([0,1])$. It corresponds
to the $x$-component of a unidirectional, zero-mean, steady incompressible
shear flow with fluid velocity field
$q(x,y)=(\alpha(y),0)~{}~{}\text{for}~{}~{}(x,y)\in\mathbb{R}\times[0,1],\quad\text{with}\quad\int_{0}^{1}\alpha(y)dy=0.$
(IBVPd) Equation (IBVPa) involves two non-dimensional parameters
$A=\frac{V}{\sqrt{\kappa/\tau}}\quad\text{and}\quad
B=\frac{\kappa\tau}{a^{2}},$ where $V$ is the characteristic speed of the
(dimensional) flow. They respectively measure the strength of advective fluid
velocity relative to the front speed (in the absence of advection), and the
square of the front thickness (in the absence of advection) to the square of
the channel width. We focus on two-dimensional straight infinite channel
domains with Neumann boundary conditions at the domain walls,
$u_{y}(x,0)=u_{y}(x,1)=0\quad\text{for}~{}~{}x\in\mathbb{R}$ (IBVPe) so that
there is no flux of $u$ across the wall of the domain. These are supplemented
by the initial conditions and far field boundary conditions, namely,
$u(x,y,0)=H(-x)~{}~{}\text{for}~{}~{}(x,y)\in\mathbb{R}\times[0,1],\quad
u(x,y,t)\to\begin{cases}1,&\text{as $x\to-\infty$}\\\ 0,&\text{as
$x\to\infty$}\end{cases}$ (IBVPf) with these limits being uniform for
$y\in[0,1]$ and $t\in[0,T]$, for any $T>0$. Here $H:\mathbb{R}\to\mathbb{R}$
is the usual Heavyside function. In what follows we will refer to the initial
boundary value problem specified by the above equations as (IBVP).

###  1.1 Background and related works

The (IBVP) was first considered by Brunet and Derrida [3] in the absence of a
background flow that is, when $A=0$. They proposed it as a model to provide
insight into the spreading of discrete systems of interacting chemical and
biological particles in homogeneous environments. They conjectured that
discreteness in concentration values can be represented by an effective cut-
off concentration $u_{c}$, where $u_{c}$ may be viewed as the effective mass
of a single particle. The idea is that for $u<u_{c}$, diffusion dominates over
growth. It is now recognised, following previous work by Tisbury and the
authors [25], that the solution to (IBVP) in the absence of a background flow
evolves at large times to a permanent form traveling wave (PTW) solution
$u(x,y,t)=U_{T}(x-vt)$ and propagation speed $v=v^{*}(u_{c})$ in the sense
that there exist functions $s:\mathbb{R}^{+}\to\mathbb{R}$ and
$U_{T}:\mathbb{R}\to\mathbb{R}$ such that

$s(t)t^{-1}\to v^{*}(u_{c})\quad\text{and}\quad u(z+s(t),y,t)\to U_{T}(z),$
(1)

as $t\to\infty$ uniformly for $z\in\mathbb{R}$, with
$(\dot{s}(t)-v^{*}(u_{c}))$ being exponentially small in $t$ as $t\to\infty$
and a similar, spatially uniform, rate of convergence in the second limit in
(1). Tisbury and the authors [24] showed that the PTW solution is, for each
$u_{c}\in(0,1)$ and KPP function $f(u)$, unique (up to translation), monotone
decreasing and positive, with $\lim_{z\to-\infty}U_{T}(z)=1$ and
$\lim_{z\to\infty}U_{T}(z)=0$. Its propagation speed $v=v^{*}(u_{c})$ is, for
fixed $u_{c}\in(0,1)$, uniquely determined by the existence of a heteroclinic
connection in the $(U,U^{\prime})$ phase plane which connects the equilibrium
point $(1,0)$, as $z\to-\infty$, to the equilibrium point $(0,0)$ as
$z\to\infty$ (the translational invariance is then fixed by requiring that
$U_{T}(0)=u_{c}$). An explicit expression for $v^{*}(u_{c})$ is not available.
However, it is straightforward to demonstrate that $v^{*}(u_{c})$ is a
continuous, monotone decreasing function of $u_{c}\in(0,1)$, with
$v^{*}(u_{c})\to 2^{-}$ as $u_{c}\to 0^{+}$ and $v^{*}(u_{c})\to 0^{+}$ as
$u_{c}\to 1^{-}$ [24] , where $2$ is the minimum propagation speed of the PTW
solution in the absence of cut-off ($u_{c}=0$). Its asymptotic form can be
approximated in the limits of $u_{c}\to 0^{+}$ and $u_{c}\to 1^{-}$. In the
first limit, Brunet and Derrida [3] predicted that the value of $v^{*}(u_{c})$
is strongly influenced by the presence of a cut-off, with

$v^{*}(u_{c})=2-\frac{\pi^{2}}{(\ln u_{c})^{2}}+O\left(\frac{1}{|\ln
u_{c}|^{3}}\right)\quad\text{as $u_{c}\to 0^{+}$},$ (2)

so that in this limit, the difference between $v^{*}(u_{c})$ and $2$ is only
logarithmically small in $u_{c}$. This behaviour was rigorously verified by
Dumortier, Popovic and Kaper [8] using geometric desingularization. Higher
order corrections were obtained by Tisbury and the authors [24] using matched
asymptotic expansions. The behaviour of $v^{*}(u_{c})$ as $u_{c}\to 0^{+}$ is
in contrast with the behaviour of $v^{*}(u_{c})$ as $u_{c}\to 1^{-}$, in which
case it was established in [24] that

$v^{*}(u_{c})=(1-u_{c})|f^{\prime}(1)|^{1/2}+O((1-u_{c})^{2}),\quad\text{as
$u_{c}\to 1^{-}$},$ (3)

and so now vanishing algebraically in $(1-u_{c})$. There are no equivalent
results for the spatially heterogenous version of (IBVP) ($A>0)$.

In this paper our aim is to understand how advection by a shear flow
influences the shape of the PTW solution and its propagation speed. This
understanding is important due to the fact that in a wide variety of
environmental and engineering applications, associated chemical or biological
reactive species are transported by fluid flows, with the simplest non-trivial
flows being shear flows. In this case, it is natural to anticipate curved PTW
solutions of the form $u(x,y,t)=U_{T}(x-vt,y)$ and propagation speed
$v=\hat{v}(A,B,u_{c})$ , where $\hat{v}(0,B,u_{c})=v^{*}(u_{c})$. The
functions $s:[0,1]\times\mathbb{R}^{+}\to\mathbb{R}$ and
$U_{T}:\mathbb{R}\times[0,1]\to\mathbb{R}$ now satisfy

$s(y,t)t^{-1}\to\hat{v}(A,B,u_{c})\quad\text{and}\quad u(z+s(y,t),y,t)\to
U_{T}(z,y),$ (4)

as $t\to\infty$ uniformly for $(z,y)\in\mathbb{R}\times[0,1]$ with
$(s_{t}(y,t)-\hat{v}(A,B,u_{c}))=o(1)$ as $t\to\infty$ uniformly in
$y\in[0,1]$. The question of existence of curved PTW solutions for KPP
reactions without cut-off (that is, $u_{c}=0$) and shear flows with bounded
cross-sections and Neumann boundary conditions was considered in detail by
Berestycki and Nirenberg [2]. They used the approach of sub- and super-
solutions on the associated semilinear elliptic boundary value problem to
establish that a unique PTW solution exists for each propagation $v\geq
v_{m}(A,B)$, where $v_{m}(A,B)$ is the (positive) minimum available
propagation speed. This PTW solution is monotone decreasing in $z$ and has
$\lim_{z\to\infty}U_{T}(z,y)=0$ and $\lim_{z\to-\infty}U_{T}(z,y)=1$,
uniformly for $y\in[0,1]$. A number of works [12, 18, 22] have shown that in
fact, for KPP reactions without cut-off, the solution to the (IBVP) approaches
that PTW with minimum speed $v_{m}(A,B)$.

The elliptic boundary value problem whose solution provides the curved PTW
presents little immediate and direct information about the quantitative
dependence of the propagation speed on the parameters and the details of the
profiles of the advective flow and reaction functions. A first
characterization of the propagation speed was derived by Gärtner and Freidlin
[13] for KPP reactions using probabilistic arguments. This variational
characterization expressed $v_{m}(A,B)$ in terms of the principal eigenvalue
of a certain linear eigenvalue problem that depends on $A$, $B$ and
$\alpha(y)$. Thus, $v_{m}(A,B)$ depends on the structure of $\alpha(y)$ but
not the detailed structure of $f(u)$ (other than it satisfying the KPP
conditions). An alternative characterization of $v_{m}(A,B)$ was subsequently
derived by Berestycki and Nirenberg [2] in terms of a quadratic linear
eigenvalue problem, where $v_{m}(A,B)$ is determined by the requirement that
the two principal eigenvalues are equal (the equivalence of the two
characterizations is shown in [28]). A variational characterization was
developed in [23] for the case of bistable and combustion type nonlinearity.

The eigenvalue problem determining the speed of propagation of the curved PTW
solution for KPP reaction functions without cut-off is readily solved
numerically. An exact analytical solution is, however, not available even for
simple shear flows. It can nevertheless be analysed in asymptotic limits of
the two parameters $A$ and $B$. For a channel width that is comparable to the
advectionless front scale thickness, i.e. when $B=O(1)$, it can be shown that
a shear flow enhances the speed of propagation, with the enhancement being
monotonic in $A$ [1]. Further asymptotic results obtained in [21] and [1]
respectively provide explicit expressions for $v_{m}(A,B)$ with

$v_{m}(A,B)=2+O(A^{2}/B)\quad\text{as}\quad A\to 0\quad\text{and}\quad
v_{m}(A,B)\sim A\alpha_{M}\quad\text{as}\quad A\to\infty,$ (5)

where the constant $\alpha_{M}\equiv\sup_{y\in[0,1]}\alpha(y)>0$ was
determined in [14]. For a channel width that is much smaller than the
advectionless front scale thickness, i.e. when $B\gg 1$, $v_{m}(A,B)$ is, at
leading order, proportional to the square root of the effective diffusivity
$\kappa_{\text{eff}}$ of the advection–diffusion problem [23] so that

$v_{m}(A,B)\sim 2\sqrt{\kappa_{\text{eff}}}\quad\text{as}\quad B\to\infty,$
(6a) uniformly for $A>0$. An explicit form for $\kappa_{\text{eff}}$ can be
obtained using homogenization (see, for example, [16, 4]) which yields
$\kappa_{\text{eff}}=1+\left(A^{2}/B\right)\langle\left(\int_{0}^{y}\alpha(y^{\prime})dy^{\prime}\right)^{2}\rangle$
(6b)

which holds uniformly for $A>0$. Here, $\left\langle
F(\cdot)\right\rangle\equiv\int_{0}^{1}F(y)dy$. In the opposite limit, when
the channel width is much larger than the advectionless front scale thickness,
i.e. when $B\ll 1$, the PTW solutions are sharp-fronted and can be
approximated by a single curve where all the reaction takes place. Majda and
Souganidis [17] (see also [12, Ch. 6]) showed that a distinguished regime
arises for $A=O(1)$ in which case the propagation speed satisfies [10, 29, 26]

$v_{m}(A,B)\sim 2+A\alpha_{M}\quad\text{as}\quad B\to 0.$ (7)

A complete description of $v_{m}(A,B)$ as $A/\sqrt{B}\to\infty$ can be readily
deduced from the analysis in [14].

A natural question is to what extent the above results apply in the presence
of a cut-off in the reaction.

###  1.2 Main results and paper structure

In this paper we consider (IBVP) in a number of natural asymptotic limits on
the parameters $A$ and $B$, with $u_{c}\in(0,1)$ fixed. One feature which we
concentrate on, amongst others, is the existence of PTW solutions, and the
detailed form of the associated propagation speed, $\hat{v}(A,B,u_{c})$,
which, for $u_{c}>0$, is unique. We use the theory of matched asymptotic
expansions to establish the limiting form of (IBVP) and/or the associated PTW
theory, and subsequently obtain expressions for $\hat{v}(A,B,u_{c})$ and the
boundary of the interface $\zeta(y)$ where $U_{T}(\zeta(y),y)=u_{c}$, in the
three natural cases of $A\to\infty$, $A\to 0$ and $A=O(1)$. Our main
conclusions are that for $A\to\infty$ with $B=O(A^{2})$, the solution to
(IBVP) is, at leading order, described by an effective equation devoid of
advection with molecular diffusivity replaced by the effective diffusivity
(6b). Thus, $\hat{v}(A,B,u_{c})$ is at leading order enhanced by the shear
flow by a factor proportional to the square root of $\kappa_{\text{eff}}$. For
$A\to 0$ we consider each of the complementary orderings $B^{-1}=o(1),B=O(1)$
and $B=o(1)$, and find that $\hat{v}(A,B,u_{c})$ is at leading order given by
$v^{*}(u_{c})$, the speed of propagation in the absence of a flow with
$\zeta(y)$ determined solely by the structure of the flow and the value of
$v^{*}(u_{c})$. The higher order correction to the speed is of $O(A)$ when
$0<B\leq O(A)$, initially increasing from zero as $B^{1/2}$, until it achieves
a maximum point located at a value $B=B_{M}(A)=O(A)$. Thereafter it is
decreasing with increasing $B$, becoming of $O(A^{2})$ for $B\geq O(1)$, and
decreasing at a rate of $B^{-1}$. At the same time, the structure of the
interface becomes increasingly deformed. Finally, for $A=O(1)$ with
$B\to\infty$, the solution to (IBVP) is, at leading order, effectively
described by the (IBVP) obtained for $A=B=0$. The situation is different for
$A=O(1)$ with $B\to 0$, when the leading order term of the propagation speed
also includes $A\alpha_{M}$, where $\alpha_{M}$ corresponds to the maximum
velocity in the channel, whilst the correction is of $O(B^{\frac{1}{2}})$. In
this case, the interface is most deformed. Contrasting our results against
(5), (6) and (7), _we conclude that the effect of a shear flow on the speed of
propagation of the PTW solution is similar with and without cut-off in the KPP
reaction, as may be anticipated_. Finally, it should be noted that in each of
the overlapping limits within and between sections 3–5, it is readily verified
that the respective asymptotic expressions for $\hat{v}(A,B,u_{c})$ match
together according to the classical Van Dyke asymptotic matching principle
[27] (see also [19, Ch. 8] for an applied analysis point of view).

The paper is structured as follows. Section 2 focuses on reformulating (IBVP)
as an equivalent moving boundary evolution problem that we refer to as (QIVP),
and as a preliminary, we examine the structure of the solution to (QIVP) at
small times. We then move on to describe two equivalent elliptic boundary
value problems that govern the existence of possible PTW solutions to (IBVP)
and (QIVP) respectively, and explore their structure as $u_{c}\to 1$. Sections
3, 4 and 5 are devoted to each of the three cases of $A\to\infty$, $A\to 0$
and $A=O(1)$ respectively. Throughout, we illustrate the theory by evaluating
expressions (either directly or through numerical integration) for two
classical shear flows: the plane Couette flow given by the linear shear

$\displaystyle\alpha(y)$ $\displaystyle=y-1/2,$ (8a) and the plane Poiseuille
flow given by $\displaystyle\alpha(y)$ $\displaystyle=-2y^{2}+2y-1/3,$ (8b)

with the constants fixed by the requirement that both of these shear flows
have mean zero flow, their half-channel mean strain rate is the same and the
maximum of the Poiseuille flow is located at the channel centre line $y=1/2$.
The paper ends with the concluding section 6.

##  2 Problem formulation

We begin this section by reformulating the parabolic evolution problem (IBVP)
as an equivalent moving boundary evolution problem that we refer to as (QIVP).
This reformulation will prove convenient at a number of stages throughout the
paper. We then move on to describe two equivalent elliptic boundary value
problems derived from (IBVP) and (QIVP) whose solution provides permanent form
travelling waves. We then consider some general results relating to this
evolution problem, and its reduction for permanent form travelling waves.

###  2.1 The moving boundary evolution problem

In this subsection we begin by developing a modification and then a
reformulation of (IBVP). Due to the discontinuity in $f_{c}(u)$ when
$u=u_{c}$, there will be a lack of full regularity in classical solutions to
(IBVP) (this is readily established, employing an argument by contradiction,
after it is shown, via the maximum principle, that any fully classical
solution to (IBVP) must have $u_{x}<0$ on $\mathbb{R}\times(0,T]$, for any
$T>0$, and so, at each $t\in(0,T]$, there exists a unique smooth curve
$x=x(y,t)$ upon which $u=u_{c}$. The contradiction then follows by choosing an
interior point on this curve, and then examining the limits in (IBVPa) as this
point is approached from left and right, under fully classical regularity and
with reaction function (IBVPb)). Therefore, for a solution to exist at all to
(IBVP), we must mildly relax regularity and the notion of fully classical
solution. Specifically, across local, spatial level curves along which
$u=u_{c}$, we admit the possibility of a _jump discontinuity in the second
spatial partial derivative of $u$ in the direction normal to such a curve_.
Furthermore, from [24], we anticipate that such a spatial curve will be
unique. With this in mind, we adopt a reformulation of (IBVP), _which under
the above restrictions_ , can be written as an equivalent moving boundary
evolution problem. First, as a measure of convenience in this reformulation,
we introduce a simple rescaling of coordinates via,

$(x,y)=(x^{\prime},L^{-1}y^{\prime})$

and the problem domain now has
$(x^{\prime},y^{\prime},t)\in\mathbb{R}\times[0,L]\times\overline{\mathbb{R}}^{+}=\bar{D}$
with $D=\mathbb{R}\times(0,L)\times\mathbb{R}^{+}$ and $L=1/\sqrt{B}.$ We next
introduce the smooth interface in $D$ as

$\mathcal{L}_{IBVP}=\\{(x^{\prime},y^{\prime},t)\in
D:x^{\prime}=s(t)+\zeta(y^{\prime},t)\\}$ (9)

where

$s\in C^{1}(\mathbb{R}^{+})\cap
C(\bar{\mathbb{R}}^{+})\quad\text{and}\quad\zeta\in
C^{1,1}([0,L]\times{\mathbb{R}}^{+})\cap C([0,L]\times\bar{\mathbb{R}}^{+}),$
(10)

and are chosen so that

$u(s(t)+\zeta(y^{\prime},t),y^{\prime},t)=u_{c}~{}~{}\forall~{}~{}(y^{\prime},t)\in[0,L]\times{\mathbb{R}}^{+},\quad\text{with}\quad\int_{0}^{L}{\zeta(y^{\prime},t)}dy^{\prime}=0~{}~{}\forall~{}~{}t\in\mathbb{R}^{+}.$
(11)

At each $t>0$, we observe that $\mathcal{L}_{IBVP}$ represents the spatial
curve for $(x^{\prime},y^{\prime})\in\mathbb{R}\times[0,L]$ on which
$u=u_{c}$. In association with $\mathcal{L}_{IBVP}$, we introduce the regions

$D_{R}=\\{(x^{\prime},y^{\prime},t):(y^{\prime},t)\in(0,L)\times{\mathbb{R}^{+}},x^{\prime}\in(s(t)+\zeta(y^{\prime},t),\infty)\\}$
(12a) and
$D_{L}=\\{(x^{\prime},y^{\prime},t):(y^{\prime},t)\in(0,L)\times\mathbb{R}^{+},x^{\prime}\in(-\infty,s(t)+\zeta(y^{\prime},t)\\}$
(12b)

with $u\geq u_{c}$ in $\bar{D}_{L}$ and $u\leq u_{c}$ in $\bar{D}_{R}$. In
this context, a classical solution will have
$u:\mathbb{R}\times[0,L]\times\overline{\mathbb{R}}^{+}\to\mathbb{R}$ having
regularity

$u\in
BC((\mathbb{R}\times[0,L]\times\overline{\mathbb{R}}^{+})\setminus(\\{0\\}\times[0,L]\times\\{0\\}))~{}\cap~{}C^{1,1,1}(\mathbb{R}\times[0,L]\times\mathbb{R}^{+})~{}\cap~{}C^{2,2,1}(D_{L}\cup
D_{R}).$ (13)

Here $BC$ represents bounded and continuous whilst, for example, $C^{1,1,1}$
represents continuous with continuous partial derivatives in the first three
arguments. The moving boundary problem is then readily formulated as follows.
First, it is convenient to introduce the mean shifted coordinate along the
channel, namely,

$\xi^{\prime}=x^{\prime}-s(t)~{}~{}\forall~{}~{}(x^{\prime},t)\in\mathbb{R}\times\overline{\mathbb{R}}^{+}.$

with the original interface now located at

$\mathcal{L}_{QIVP}=\\{(\xi^{\prime},y^{\prime},t):\xi^{\prime}=\zeta(y^{\prime},t),~{}(y^{\prime},t)\in(0,L)\times\mathbb{R}^{+}\\}.$
(14)

At each $t>0$, we note that a spatial normal to $\mathcal{L}_{QIVP}$, pointing
left to right, is given by

$\textbf{n}(\zeta(y^{\prime},t),y^{\prime})=(1,-\zeta_{y^{\prime}}(y^{\prime},t))~{}~{}\forall~{}~{}(y^{\prime},t)\in(0,L)\times\mathbb{R}^{+}.$
$u_{t}+(A\bar{\alpha}(y^{\prime})-\dot{s}(t))u_{\xi^{\prime}}=\nabla^{2}u+f_{c}(u),~{}~{}(\xi^{\prime},y^{\prime},t)\in
Q_{L}\cup Q_{R},$ (QIVPa) $u\geq u_{c}~{}\text{in}~{}Q_{L},~{}~{}u\leq
u_{c}~{}\text{in}~{}Q_{R},$ (QIVPb)
$u(\xi^{\prime},y^{\prime},0)=H(-\xi^{\prime}),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L],$
(QIVPc)
$u(\xi^{\prime},y^{\prime},t)\to\begin{cases}1~{}~{}\text{as}~{}~{}\xi^{\prime}\to-\infty,\\\
0~{}~{}\text{as}~{}~{}\xi^{\prime}\to\infty,\end{cases}$ (QIVPd) uniformly for
$(y^{\prime},t)\in[0,L]\times[0,T]$, any $T>0$,
$u_{y^{\prime}}(\xi^{\prime},0,t)=u_{y^{\prime}}(\xi^{\prime},L,t)=0,~{}~{}(\xi^{\prime},t)\in\mathbb{R}\times\overline{\mathbb{R}}^{+},$
(QIVPe) and at the moving boundary, continuity of $u$ and its two spatial
partial derivatives requires (with a direct calculation of directional
derivatives),
$u(\zeta(y^{\prime},t)^{+},y^{\prime},t)=u(\zeta(y^{\prime},t)^{-},y^{\prime},t)=u_{c},~{}~{}(y^{\prime},t)\in[0,L]\times\mathbb{R}^{+},$
(QIVPf) $s(0)=0,~{}\zeta(y^{\prime},0)=0,~{}~{}y^{\prime}\in[0,L],$ (QIVPg)
$[\nabla
u\cdot\bm{n}(\zeta(y^{\prime},t),y^{\prime},t))]_{L}^{R}=0,~{}~{}(y^{\prime},t)\in(0,L)\times\mathbb{R}^{+}.$
(QIVPh) In the above, $\nabla=(\partial_{\xi^{\prime}},\partial_{y^{\prime}})$
is the usual gradient operator in the coordinates $(\xi^{\prime},y^{\prime})$,
and,
$\begin{split}Q_{R}&=\\{(\xi^{\prime},y^{\prime},t):(y^{\prime},t)\in(0,L)\times\mathbb{R}^{+},~{}\xi^{\prime}\in(\zeta(y^{\prime},t),\infty)\\},\\\
Q_{L}&=\\{(\xi^{\prime},y^{\prime},t):(y^{\prime},t)\in(0,L)\times\mathbb{R}^{+},~{}\xi^{\prime}\in(-\infty,\zeta(y^{\prime},t))\\},\end{split}$
(QIVPi) with $[\cdot]_{L}^{R}$ indicating the difference in limits across
$\mathcal{L}_{QIVP}$ from $Q_{L}$ to $Q_{R}$. In addition,
$\bar{\alpha}(y^{\prime})=\alpha(y^{\prime}/L)~{}~{}\forall~{}~{}y^{\prime}\in[0,L]$
with, from (IBVPd) and (11),
$\int_{0}^{L}{\bar{\alpha}(y^{\prime})}dy^{\prime}=0,\quad\text{and}\quad\int_{0}^{L}{\zeta(y^{\prime},t)}dy^{\prime}=0~{}~{}\forall~{}~{}t\in\mathbb{R}^{+}.$
(QIVPj)

The regularity condition given in (13) becomes,

$u\in
BC((\mathbb{R}\times[0,L]\times[0,\infty))\setminus(\\{0\\}\times[0,L]\times\\{0\\}))~{}\cap~{}C^{1,1,1}(\mathbb{R}\times[0,L]\times(0,\infty))~{}\cap~{}C^{2,2,1}(Q_{L}\cup
Q_{R}).$ (15)

It is useful to observe that on using the regularity requirements on $u$ as
given in (15), together with the earlier regularity requirements on $\zeta$
and $s$ (10), we can readily establish that $u_{\xi^{\prime}\xi^{\prime}}$,
$u_{y^{\prime}y^{\prime}}$ and $u_{\xi^{\prime}y^{\prime}}$ each have finite
limits from $Q_{L}$ and $Q_{R}$ as each point on $\mathcal{L}_{QIVP}$ is
approached. In addition, on using the translation invariance of (QIVP) in the
coordinate $\xi^{\prime}$, together with application of the classical
parabolic strong maximum principle and comparison theorem (in both $Q_{L}$ and
$Q_{R}$ respectively) we are able to establish the following basic properties
for (QIVP), namely,

$0<u(\xi^{\prime},y^{\prime},t)<u_{c}~{}~{}\text{in}~{}~{}Q_{R},\quad
u_{c}<u(\xi^{\prime},y^{\prime},t)<1~{}~{}\text{in}~{}~{}Q_{L},~{}\forall
t>0,$ (16)

and,

$u(\xi^{\prime},y^{\prime},t)~{}~{}\text{is strictly monotone decreasing
with}~{}~{}\xi^{\prime}\in\mathbb{R}~{}~{}\text{for
fixed}~{}(y^{\prime},t)\in(0,L)\times(0,\infty).$ (17)

Use of the regularity conditions on $u$ and $\zeta$, together with the moving
boundary conditions (QIVPf) and (QIVPh), the chain rule, and the PDE (QIVPa),
in both $Q_{L}$ and $Q_{R}$, it is also readily established (using continuity
of the two spatial partial derivatives $u_{ss}$ and $u_{ns}$ across the
interface, together with (QIVPa), when written in terms of tangential-normal
coordinates $(s,n)$ at the interface $\mathcal{L}_{QIVP}$) that

$[u_{nn}]_{L}^{R}=f_{c},$ (18)

at all points on $\mathcal{L}_{QIVP}$, with $u_{nn}$ being the second spatial
derivative of $u$ with respect to distance in the direction of $\textbf{n},$
as the point on $\mathcal{L}_{QIVP}$ is approached.

We are now able to construct, via the method of matched asymptotic expansions,
a solution to (QIVP) as $t\to 0^{+}$ which gives a uniform approximation to
$u$ throughout the spatial domain. In particular, this analysis provides
information regarding the early dynamics of the interface in (QIVP), and the
relative importance of advection, diffusion and reaction in the dynamics. We
note that the ability to construct such an approximation gives a strong
indication that (QIVP) is, at least locally in $t$, well-posed. The details of
this construction are extensive and as such are relegated to the Appendix.

Finally we consider, at a given $t=O(1)^{+}$, a local analysis close to the
point of intersection of the spatial moving boundary and the channel boundary,
at $(\xi^{\prime},y^{\prime})=(\zeta(0,t),0)$111In this context, for any
variable $\lambda$, we will henceforth write $\lambda=O(1)>0$ as
$\lambda=O(1)^{+}$, and correspondingly, $\lambda=O(1)<0$ as
$\lambda=O(1)^{-}$.. After some calculation, it follows from (QIVPa),
(QIVPb),(QIVPe) and (QIVPf) that,

$u(r,\theta,t)=\begin{cases}u_{c}-A_{+}(t)r^{\frac{\pi}{2\alpha(t)}}\text{cos}(\frac{\pi\theta}{2\alpha(t)})+o(r^{\frac{\pi}{2\alpha(t)}})~{}~{}\text{as}~{}~{}r\to
0^{+}~{}~{}\text{in}~{}~{}\bar{Q}_{+},\\\
u_{c}+A_{-}(t)r^{\frac{\pi}{2(\pi-\alpha(t))}}\text{cos}(\frac{\pi(\pi-\theta)}{2(\pi-\alpha(t))})+o(r^{\frac{\pi}{2(\pi-\alpha(t))}})~{}~{}\text{as}~{}~{}r\to
0^{+}~{}~{}\text{in}~{}~{}\bar{Q}_{-},\end{cases}$ (19)

Here $A_{+}(t)$ and $A_{-}(t)$ are positive (via (16)), smooth, globally
determined functions for $t=O(1)^{+}$, whilst $(r,\theta)$ are plane polar
coordinates, given by

$\xi^{\prime}=\zeta(0,t)+r\text{cos}\theta,~{}~{}y^{\prime}=r\text{sin}\theta,$
(20a) with $r\geq 0$ and $0\leq\theta\leq\pi$. In addition,
$\text{tan}\alpha(t)=\zeta_{y^{\prime}}(0,t)~{}~{}t>0,$ (20b)

so that, locally,

$\bar{Q}_{+}=\\{(r,\theta):r\geq
0,~{}~{}0\leq\theta\leq\alpha(t)\\},\quad\bar{Q}_{-}=\\{(r,\theta):r\geq
0,~{}~{}\alpha(t)\leq\theta\leq\pi\\},$ (21)

with

$0<\alpha(t)<\pi$ (22)

at each $t=O(1)^{+}$. It remains to apply condition (QIVPh), which finally
requires that,

$\alpha(t)=\frac{1}{2}\pi~{}~{}\text{and}~{}~{}A_{+}(t)=A_{-}(t).$ (23)

A similar analysis follows close to the intersection point of the spatial
moving boundary and the channel boundary at
$(\xi^{\prime},y^{\prime})=(\zeta(L,t),L).$ We observe from (23) that the
spatial moving boundary makes normal contact with the channel boundary at each
$t>0$. It should be noted that the initial condition (QIVPc) requires that
$\zeta(0,t)\to 0$ as $t\to 0^{+}$. In addition, the structure of the solution
to (QIVP) when $t$ is small, which is developed in the Appendix, does in fact
allow us to obtain some additional information regarding the indeterminate
functions $A_{\pm}(t)$. In particular, asymptotic matching with particular
boundary regions, which are located, respectively when $y^{\prime}=O(t^{1/2})$
and $y^{\prime}=L-O(t^{1/2})$ (labelled as regions $\mathrm{I}_{L,R}$ in the
Appendix) determines (via Van Dyke’s matching principle) that

$A_{\pm}(t)=O(t^{-\frac{1}{2}})~{}~{}\text{as}~{}~{}t\to 0^{+}.$ (24)

This condition, together with (23), determines that _when $t$ is small_, the
spatial region of validity of the expansion (19), local to the free boundary
contact point with the wall at $y^{\prime}=0$, requires the restriction $0\leq
r\ll t^{\frac{1}{2}}$ with $0\leq\theta\leq\pi$.

We next consider permanent form travelling front structures which may develop
as large-$t$ attractors in the solution to (IBVP), and equivalently (QIVP).

###  2.2 Permanent form travelling fronts

We anticipate that as $t\to\infty$ a permanent form travelling front structure
will develop in the solution to (IBVP) (and equivalently (QIVP)), advancing
with a non-negative, constant propagation speed, and allowing for the
transition from the unreacted state $u=0$ to the fully reacted state $u=1$.
This anticipation is supported by the theory developed in the earlier papers
by Tisbury and the authors, [24] and [25], in the case when the shear flow is
absent. With this in mind, we begin by formulating the elliptic boundary value
problems (which are equivalent) associated with permanent form travelling
front solutions in both the formulations (IBVP) and (QIVP). As will be seen in
the later sections, one or other of these equivalent formulations usually
presents itself as the most natural to work with in each limiting case that we
consider. In what follows, we will refer to a permanent form travelling front
solution to either of (IBVP) or (QIVP) as a PTW solution.

A PTW solution, with propagation speed $v\geq 0$, is a non-negative solution
to (IBVP), which depends only upon $y$ and the travelling coordinate,

$z=x-vt.$

Thus, $U_{T}:\mathbb{R}\times[0,1]\to\mathbb{R}$ is a PTW soluton to (IBVP),
with propagation speed $v\geq 0$, when,

$U_{Tzz}+BU_{Tyy}+(v-A\alpha(y))U_{Tz}+f_{c}(U_{T})=0,~{}~{}(z,y)\in\mathbb{R}\times(0,1),$
(BVPa) with $U_{T}\in C^{1}(\mathbb{R}\times[0,1])\cap
C^{2}(\mathbb{R}\times((0,1)\setminus\mathcal{L}_{BVP}))$, where the interface
is now, $\mathcal{L}_{BVP}=\\{(z,y)\in\mathbb{R}\times(0,1):u(z,y)=u_{c}\\}.$
(BVPb) In addition, $U_{T}(z,y)\geq
0~{}~{}\forall~{}~{}(z,y)\in\mathbb{R}\times[0,1],$ (BVPc)
$U_{Ty}(z,0)=U_{Ty}(z,1)=0~{}~{}\forall~{}~{}z\in\mathbb{R},$ (BVPd)
$U_{T}(z,y)\to\begin{cases}0~{}~{}\text{as}~{}~{}z\to\infty,\\\
1~{}~{}\text{as}~{}~{}z\to{-\infty}\end{cases}$ (BVPe)

uniformly for $y\in[0,1]$. This elliptic boundary value problem, which we
henceforth refer to as (BVP), may be regarded as a nonlinear eigenvalue
problem, with eigenvalue $v\geq 0$. Now let
$U_{T}:\mathbb{R}\times[0,1]\to\mathbb{R}$ be a PTW with propagation speed
$v\geq 0$. We observe that, on any closed bounded interval
$I\subset\mathbb{R}$, then,

$f_{c}(\lambda)\lambda^{-1}~{}\text{is bounded}~{}~{}\forall~{}~{}\lambda\in
I,$ (25)

via (IBVPb) and (IBVPc). It then follows from (BVPa), (BVPd) and (BVPe),
together with the strong elliptic maximum principle (see, for example, Ch. 3
in [9]), that,

$U_{T}(z,y)>0~{}~{}\forall~{}~{}(z,y)\in\mathbb{R}\times[0,1].$ (26)

A similar argument, using (BVPa), (BVPd) and (BVPe), with (IBVPb), (IBVPc),
and the strong elliptic maximum principle establishes that

$U_{T}(z,y)<1~{}~{}\forall~{}~{}(z,y)\in\mathbb{R}\times[0,1].$ (27)

With the moving boundary reformulation in mind, we return to the formulation
of (QIVP) which determines that $\mathcal{L}_{BVP}$ can be represented in the
following form,

$\mathcal{L}_{BVP}=\\{(z,y):z=\gamma(y)~{}~{}\text{for}~{}~{}y\in[0,1]~{}~{}\text{with}~{}~{}U_{T}(\gamma(y),y)=u_{c}\\}$
(28)

with (via the regularity required on
$U_{T}:\mathbb{R}\times[0,1]\to\mathbb{R}$) $\gamma\in C^{1}([0,1])$, and
$u\geq u_{c}$ to the left of $\mathcal{L}_{BVP}$ and $u\leq u_{c}$ to the
right of $\mathcal{L}_{BVP}$, respectively. With this structure, we can then
apply the strong maximum and minimum principle, in each of the regions to the
right and left of $\mathcal{L}_{BVP}$, to establish that,

$U_{Tz}(z,y)<0~{}~{}\forall~{}~{}(z,y)\in\mathbb{R}\times(0,1)\quad\text{with}\quad
U_{T}(z,y)\begin{cases}>u_{c}~{}~{}\text{to left
of}~{}~{}\mathcal{L}_{BVP},\\\ <u_{c}~{}~{}\text{to right
of}~{}~{}\mathcal{L}_{BVP}.\end{cases}$ (29)

The problem (BVP) is translation invariant in the coordinate $z$, and we will
fix this invariance by henceforth requiring that

$\int_{0}^{1}{\gamma(y)}dy=0.$ (30)

We now consider the equivalent moving boundary formulation for a PTW solution.
With translational invariance fixed through (30), then a PTW is simply a
_steady solution_ of (QIVP), and so the spatial moving boundary in (QIVP) for
a PTW, with propagation speed $v\geq 0$, is now located at,

$\mathcal{L}_{QBVP}=\\{(\xi^{\prime},y^{\prime}):~{}~{}\xi^{\prime}=\zeta(y^{\prime}),~{}~{}y^{\prime}\in[0,L]\\}$
(31)

where, as the solution is now steady, the dependence on $t$ has been omitted,
whilst in relation to (BVP),

$\xi^{\prime}=z,~{}~{}\zeta(y^{\prime})=\gamma(y^{\prime}L^{-1})$ (32)

for each $y^{\prime}\in[0,L]$, with $y$ and $y^{\prime}$ related as before,
whilst translational invariance (30) becomes (following (11) and (QIVPj))

$\int_{0}^{L}{\zeta(y^{\prime})}dy^{\prime}=0$ (33)

with

$\dot{s}(t)=v.$ (34)

The (BVP) formulation becomes the equivalent moving boundary formulation that
we henceforth refer to as (QBVP):

$\nabla^{2}U_{T}-(A\bar{\alpha}(y^{\prime})-v)U_{T\xi^{\prime}}+f_{c}(U_{T})=0,~{}~{}(\xi^{\prime},y^{\prime})\in
Q_{L}\cup Q_{R},$ (QBVPa) with, $U_{T}\geq
u_{c}~{}\text{in}~{}Q_{L},~{}~{}U_{T}\leq u_{c}~{}\text{in}~{}Q_{R},$ (QBVPb)
and
$U_{T}(\xi^{\prime},y^{\prime})\to\begin{cases}1~{}~{}\text{as}~{}~{}\xi^{\prime}\to-\infty,\\\
0~{}~{}\text{as}~{}~{}\xi^{\prime}\to\infty,\end{cases}$ (QBVPc) uniformly for
$y^{\prime}\in[0,L]$, whilst
$U_{Ty^{\prime}}(\xi^{\prime},0)=U_{Ty^{\prime}}(\xi^{\prime},L)=0,~{}~{}\xi^{\prime}\in\mathbb{R}.$
(QBVPd) At the boundary $\mathcal{L}_{QBVP}$,
$U_{T}(\zeta(y^{\prime})^{+},y^{\prime})=U_{T}(\zeta(y^{\prime})^{-},y^{\prime})=u_{c},~{}~{}y^{\prime}\in[0,L],$
(QBVPe) and $[\nabla
U_{T}\cdot\bm{n}(\zeta(y^{\prime}),y^{\prime}))]_{L}^{R}=0,~{}~{}y^{\prime}\in(0,L).$
(QBVPf)

Throughout the rest of the paper, we will consider PTW solutions via either
(BVP) or (QBVP), depending upon convenience. Before moving on to detailed
analysis, we first make some preliminary observations regarding PTW solutions.
Following the theory developed in [24], for the case when
$\bar{\alpha}(y^{\prime})=0$ for $y^{\prime}\in[0,L]$, we anticipate that, for
a _given_ shear flow profile $\bar{\alpha}(y^{\prime})$, then (QBVP) (and
equivalently (BVP)) has a PTW solution if and only if

$v=\hat{v}(A,B,u_{c})>0$ (35)

with $\hat{v}:(0,\infty)\times\mathbb{R}^{+}\times(0,1)\to\mathbb{R}$ a
continuous function, and that this PTW solution is unique. We will see that
this is supported by the asymptotic results developed in the following
sections. In addition, following [24], we anticipate that,

$\hat{v}(A,B,u_{c})\to 0~{}~{}\text{as}~{}~{}u_{c}\to 1$ (36)

whilst,

$\hat{v}(A,B,u_{c})\to v_{m}(A,B)~{}~{}\text{as}~{}~{}u_{c}\to 0$ (37)

where, following [2] $v_{m}(A,B)>0$ is the _minimum propagation speed for PTW
solutions to the corresponding KPP problem in the absence of cut-off_.

###  2.3 Asymptotic structure of the solution to (QBVP) as $\bm{u_{c}\to 1}$.

To complete this section we consider in more detail the form of
$\hat{v}(A,B,u_{c})$ as $u_{c}\to 1$ with $A,B=O(1)$. For this purpose, it is
convenient to use the moving boundary formulation for PTW solutions, namely
(QBVP). We integrate equation (QBVPa) over $Q_{L}$ and $Q_{R}$ and apply
Green’s Theorem in the plane, using conditions (QIVPj), (QBVPb) – (QBVPf) and
taking care with the existence of the associated improper integrals, which
gives, in general,

$\hat{v}(A,B,u_{c})=\frac{1}{L}\iint_{\bar{Q}_{L}}\
f_{c}(U_{T}(\xi^{\prime},y^{\prime}))\,d\xi^{\prime}\,dy^{\prime}.$ (38)

We may now restrict attention to the spatial subdomain $Q_{L}$. In $Q_{L}$ it
follows directly from (QIVPf) and (17) that $U_{T}=1-O((1-u_{c}))$ whilst a
balancing of terms in the PDE (QIVPa) in $Q_{L}$ determines that
$\hat{v}=O((1-u_{c}))$, as $u_{c}\to 1^{-}$. Therefore we expand in $Q_{L}$
as,

$\hat{v}(A,B,u_{c})=(1-u_{c})\bar{v}(A,B)+o((1-u_{c})),$ (39)
$U_{T}(\xi^{\prime},y^{\prime})=1-(1-u_{c})\bar{u}(\xi^{\prime},y^{\prime})+o((1-u_{c})),$
(40)

with $(\xi^{\prime},y^{\prime})\in\bar{Q}_{L}$, and $A,B=O(1)$. After some
straightforward calculation, we find that,

$\bar{u}(\xi^{\prime},y^{\prime})=c\tilde{\phi}(y^{\prime})e^{\lambda_{0}\xi^{\prime}},~{}~{}(\xi^{\prime},y^{\prime})\in\bar{Q}_{L},$
(41)

with $\lambda=\lambda_{0}>0$ being the smallest positive eigenvalue _with a
principal eigenfunction_ , for the quadratic linear eigenvalue problem,

$\tilde{\phi}^{\prime\prime}+(\lambda^{2}-A{\bar{\alpha}}(y^{\prime})\lambda+f_{c}^{\prime}(1))\tilde{\phi}=0,~{}~{}y^{\prime}\in(0,L),$
(42) $\tilde{\phi}^{\prime}(0)=\tilde{\phi}^{\prime}(L)=0,$ (43)
$\tilde{\phi}(y^{\prime})>0~{}~{}\forall~{}~{}y^{\prime}\in[0,L]$ (44)

with normalisation,

$\int_{0}^{L}{\tilde{\phi}(y^{\prime})}dy^{\prime}=1,$ (45)

and where $c$ is a constant to be determined (it is readily established that,
with $f_{c}^{\prime}(1)<0,$ this quadratic linear eigenvalue problem has
exactly two distinct eigenvalues which have principal eigenfunctions, and
these two eigenvalues are real, and have opposite sign). We now determine
$\zeta(y^{\prime})$ using (40) and (41) in the remaining boundary condition
(QBVPe) to obtain,

$\zeta(y^{\prime})=-\frac{1}{\lambda_{0}}\log{(c\tilde{\phi}(y^{\prime}))}$
(46)

for $y^{\prime}\in[0,L]$. It now remains to apply the final condition (33)
after which we arrive at,

$c=e^{-M}\quad\text{with}\quad
M=\langle\log{\tilde{\phi}}\rangle=\frac{1}{L}\int_{0}^{L}{\log{\tilde{\phi}(s)}}ds.$
(47)

Thus, the moving boundary is located at,

$\xi^{\prime}=\frac{1}{\lambda_{0}}(\langle\log{\tilde{\phi}}\rangle-\log{\tilde{\phi}(y^{\prime})})+o(1),$
(48)

as $u_{c}\to 1^{-}$ with $y^{\prime}\in[0,L]$. Also, from (38) – (41), with
(47), we obtain

$\bar{v}(A,B)=\frac{1}{L}c|f_{c}^{\prime}(1)|\iint_{\bar{Q}_{L}}\
\tilde{\phi}(y^{\prime})e^{\lambda_{0}\xi^{\prime}}\,d\xi^{\prime}\,dy^{\prime},$
(49)

recalling that both $\lambda_{0}$ and $\tilde{\phi}$ depend upon $A$ and $B$.
The double integral can be evaluated directly, using (47) and (48), as having
value $\frac{L}{\lambda_{0}c}$. Thus we finally obtain,

$\bar{v}(A,B)=\frac{|f_{c}^{\prime}(1)|}{\lambda_{0}(A,B)}.$ (50)

Clearly, in the absence of advection (corresponding to $A=0$) a direct
calculation gives $\lambda_{0}(0,B)=\sqrt{|f_{c}^{\prime}(1)|}$. In this case,
equation (50) recovers the asymptotic result obtained in [24]. In addition, it
is readily established that

$\lambda_{0}(A,B)=H(I(A,B))~{}~{}\forall~{}~{}(A,B)\in\overline{\mathbb{R}}^{+}\times\mathbb{R}^{+}$
(51)

where $H:\mathbb{R}\to\mathbb{R}$ is the _positive-valued and strictly
monotone increasing_ function

$H(X)=\frac{1}{2}\left(X+\sqrt{(X^{2}+4|f^{\prime}_{c}(1)|)}\right)~{}~{}\forall~{}~{}X\in\mathbb{R}$

and

$I(A,B)=A\int_{0}^{L}{\overline{\alpha}(s)\overline{\phi}(s)}ds~{}~{}\forall~{}~{}(A,B)\in\overline{\mathbb{R}}^{+}\times\mathbb{R}^{+}$

recalling that $B=L^{-2}$. We now set

$m(B)=\text{inf}_{s\in[0,L]}\overline{\alpha}(s)<0\quad\text{and}\quad
M(B)=\text{sup}_{s\in[0,L]}\overline{\alpha}(s)>0.$

It then follows from (44) and (45) that

$Am(B)<I(A,B)<AM(B)~{}~{}\forall~{}~{}(A,B)\in\overline{\mathbb{R}}^{+}\times\mathbb{R}^{+}$

after which we obtain, via (51) and monotonicity of $H(\cdot)$, that

$H(Am(B))<\lambda_{0}(A,B)<H(AM(B))~{}~{}\forall~{}~{}(A,B)\in\overline{\mathbb{R}}^{+}\times\mathbb{R}^{+}.$
(52)

We observe that,

$H(Am(B))<\lambda_{0}(0,B)=\sqrt{|f^{\prime}_{c}(1)|}<H(AM(B))~{}~{}\forall~{}~{}(A,B)\in\mathbb{R}^{+}\times\mathbb{R}^{+}.$

Thus, although the bounds (52) constrain (via (50)) the propagation speed as
$u_{c}\to 1^{-}$, they do not indicate whether the propagation speed is
enhanced or retarded by the inclusion of advection.

Figure 1: A graph of (left) the principal eigenvalue $\lambda_{0}(A,B)$ to the
quadratic linear eigenvalue problem (42)–(43) and (right) $\bar{v}(A,B)$
determined for $f^{\prime}_{c}(1)=-1$ using (50). Results are obtained
numerically for the Couette (thick solid lines) and the Poiseuille flow (thin
solid lines) for (left) $B=0.1$ (bottom), $1$ (middle) and $10$ (top) and in
reverse order for (right).

We now compute the positive principal eigenvalue $\lambda_{0}(A,B)$ to the
quadratic linear eigenvalue problem for the Couette flow (8a) and the
Poiseuille flow (8b) and use (50) with $f^{\prime}_{c}(1)=-1$ to evaluate
$\bar{v}(A,B)$ in these two example cases. We use a standard second-order
finite-difference discretization of (42)–(43). The resulting matrix eigenvalue
problem is solved for a range of values of $A$ and $B$ using MATLAB’s routine
polyeigs. We choose the spatial resolution $h$ to satisfy $1/h=200$. The
results are shown in Figure 1. It is clear that for these particular flows,
the propagation speed is enhanced by advection, increasing with $A$ and
decreasing with $B$. Furthermore, for all values of $A$ and $B$ considered,
the propagation speed in the Couette flow remains larger than in the plane
Poiseuille flow.

We now proceed to consider (IBVP) or, equivalently (QIVP) and (BVP) or,
equivalently (QBVP), in a number of significant asymptotic limits in the two
key parameters $(A,B)$.

##  3 Slowly varying front with strong advection: $B^{-1}=o(1)$ as
$A\to\infty$

In this section we consider (IBVP) when the advection velocity scale is large
compared to the advectionless front propagation speed, and the channel width
is small relative to the advectionless front scale thickness. In terms of the
two parameters $A$ and $B$, we can formalise this by considering $A\gg 1$ and
$B\gg 1$, with the principal balances in the PDE (IBVPa) leading us to
consider the limit $A\to\infty$, with $B=O(A^{2})$ as $A\to\infty$. For ease
of notation, we write

$A=\epsilon^{-1}~{}~{}\text{and}~{}~{}B=\bar{B}\epsilon^{-2}$ (53)

with $\bar{B}=O(1)^{+}$ as $\epsilon\to 0$. In what follows, to avoid issues
with the discontinuity in the cut-off nonlinearity, it proves convenient to
make use of an exact result, obtained via (IBVPa), (IBVPe) and regularity on
$u$, and given by,

$\displaystyle\left(\int_{0}^{1}{u(x,s,t)}ds\right)_{t}+\epsilon^{-1}\left(\int_{0}^{1}{\alpha(s)u(x,s,t)}ds\right)_{x}=\left(\int_{0}^{1}{u(x,s,t)}ds\right)_{xx}+\int_{0}^{1}{f_{c}(u(x,s,t))}ds$
(54) $\displaystyle~{}~{}\forall~{}~{}(x,t)\in\mathbb{R}\times\mathbb{R}^{+}.$

Moreover, since the cut-off nonlinearity does not appear, at least up to order
$O(\epsilon)$, we are able to work, to our advantage, directly with (IBVP),
rather than the moving boundary formulation. We now introduce the expansion,

$u(x,y,t;\epsilon)=u_{0}(x,y,t)+\epsilon u_{1}(x,y,t)+O(\epsilon^{2})$ (55)

as $\epsilon\to 0$, with $(x,y,t)\in\mathbb{R}\times[0,1]\times[0,\infty)$. On
substitution from (55) into (IBVP), we obtain, at leading order, the following
problem for $u_{0}(x,y,t)$, namely,

$u_{0yy}=0,~{}~{}(x,y,t)\in\mathbb{R}\times(0,1)\times(0,\infty),$ (56a) which
must be solved subject to
$u_{0y}=0,~{}~{}(x,y,t)\in\mathbb{R}\times\\{0,1\\}\times(0,\infty),$ (56b)

which requires,

$u_{0}(x,y,t)=\bar{u}(x,t),~{}~{}(x,y,t)\in\mathbb{R}\times[0,1]\times[0,\infty),$
(57)

with $\bar{u}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ to be determined
subject to suitable regularity, with

$\bar{u}(x,t)\to\begin{cases}1~{}~{}\text{as}~{}~{}x\to-\infty,\\\
0~{}~{}\text{as}~{}~{}x\to\infty,\end{cases}$ (58)

uniformly for $t\in[0,T]$, any $T>0$, and

$\bar{u}(x,0)=H(-x),~{}~{}x\in\mathbb{R}.$ (59)

At $O(\epsilon)$ in (IBVP) we obtain from (IBVPa)

$u_{1yy}=\frac{1}{\bar{B}}\alpha(y)\bar{u}_{x}(x,t),~{}~{}~{}~{}(x,y,t)\in\mathbb{R}\times(0,1)\times(0,\infty),$
(60a) which must be solved subject to
$u_{1y}=0,~{}~{}(x,y,t)\in\mathbb{R}\times\\{0,1\\}\times(0,\infty).$ (60b)

The solution to (60) is given by,

$u_{1}(x,y,t)=\bar{V}(x,t)+\frac{1}{\bar{B}}\widetilde{\alpha}(y)\bar{u}_{x}(x,t)~{}~{}~{}~{}(x,y,t)\in\mathbb{R}\times[0,1]\times(0,\infty).$
(61)

Here $\bar{V}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ is to be determined,
with suitable regularity, and

$\widetilde{\alpha}(y)=\int_{0}^{y}\left(\int_{0}^{\mu}{\alpha(s)}ds\right)d\mu~{}~{}\forall~{}~{}y\in[0,1].$
(62)

We observe from (62) that

$\widetilde{\alpha}^{\prime}(y)=\int_{0}^{y}{\alpha(s)}ds,~{}~{}y\in[0,1],$
(63)

so that, $\widetilde{\alpha}^{\prime}(0)=\widetilde{\alpha}^{\prime}(1)=0$ via
(IBVPd). We now substitute from (55), (57), (61) and (62) into (54), and
perform the integrations to obtain, at $O(1)$, the following PDE for
$\bar{u}(x,t)$, namely,

$\bar{u}_{t}=(1+\bar{B}^{-1}\Delta)\bar{u}_{xx}+f_{c}(\bar{u}),~{}~{}(x,t)\in\mathbb{R}\times(0,\infty),$
(64a) with $\Delta=\int_{0}^{1}{(\widetilde{\alpha}^{\prime}(s))^{2}}ds>0$
(64b)

when $\alpha(y)$ is nontrivial. Equation (64a) must be solved subject to the
initial and boundary conditions (59) and (58). With the scaling transformation
$X=(1+\bar{B}^{-1}\Delta)^{-\frac{1}{2}}x$. the initial value problem (64a),
(59) and (58) for $\bar{u}$ reduces precisely to that studied in detail
recently in [24, 25]. In [24] it is established that the PDE (64) has a unique
PTW solution with propagation speed

$v=\sqrt{(1+\bar{B}^{-1}\Delta)}v^{\ast}(u_{c}),$ (65)

with $v^{\ast}(u_{c})$ as given in [24]. It should be noted that
$1+\bar{B}^{-1}\Delta=\kappa_{\text{eff}}$, the effective diffusivity
introduced in subsection 1.1. The initial boundary value problem (64a), (59)
and (58) is studied in detail in [25], where it is principally established
that $\bar{u}$ evolves into a PTW structure as $t\to\infty$. We are now able
to interpret this for (IBVP) via (55). At _leading order_ , as $A\to\infty$
with $\bar{B}=(B/A^{2})=O(1)$, we observe that $u$ becomes rapidly homogeneous
in $y$ (on a time scale $t=O(A^{-2})$ as $A\to\infty$) and thereafter is
governed by the one-dimensional cut-off KPP reaction-diffusion equation (64a).
The effect of the advective shear flow is simply to enhance the streamwise
diffusion coefficient from unity to $(1+(A^{2}/B)\Delta)$. For each $A$, $B$,
a unique PTW exists, which has propagation speed,

$\hat{v}(A,B,u_{c})=(1+\bar{B}^{-1}(A,B)\Delta)^{\frac{1}{2}}v^{\ast}(u_{c})+o(1)$
(66)

as $A\to\infty$, with $\bar{B}=B/A^{2}=O(1)$ and $v^{\ast}(u_{c})$ as given in
[24]. This PTW solution forms the large-$t$ attractor for (IBVP).

As $B$ reduces in order relative to $A=\epsilon^{-1}$, an examination of the
PDE (IBVPa) reveals that the balancing of terms at $O(\epsilon)$ undergoes a
change, in particular when the distinguished limit becomes
$B=O(\epsilon^{-\frac{2}{n}})$ with $n>1$ . In this case we write

$B=\bar{B}\epsilon^{-\frac{2}{n}}$ (67)

with, again, $\bar{B}=O(1)$ as $\epsilon\to 0$. Returning to PDE (IBVPa), the
natural scale in the $x$ coordinate is now of $O(\epsilon^{-(1-\frac{1}{n})})$
(so that the front thickness is increasing with $n$). Thus we introduce the
coordinate $X$, with

$x=\bar{B}^{\frac{1}{2}(n-1)}\epsilon^{-(1-\frac{1}{n})}X$ (68)

and the $O(1)$ factor is for algebraic convenience. We next follow the earlier
case, when $n=1$, and expand as,

$u(X,y,t;\epsilon)=\bar{u}(X,t)+O(\epsilon^{\frac{1}{n}})$ (69)

as $\epsilon\to 0$ with $(X,y,t)\in\mathbb{R}\times[0,1]\times[0,\infty)$.
Here $\bar{u}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ now satisfies,

$\bar{u}_{t}=\bar{B}^{-n}\Delta\bar{u}_{XX}+f_{c}(\bar{u}),~{}~{}(X,t)\in\mathbb{R}\times(0,\infty),$
(70)

with suitable regularity, and again subject to initial and boundary conditions
(59) and (58).The conclusions regarding (IBVP) and (BVP) are thus as for the
case $n=1$, _except_ now the front is on the stretched length scale
$x=O(A^{(1-\frac{1}{n})})$ and (66) is modified to,

$\hat{v}(A,B,u_{c})=\frac{\sqrt{\Delta}}{\sqrt{\bar{B}(A,B)}}A^{(1-\frac{1}{n})}v^{\ast}(u_{c})+o(A^{(1-\frac{1}{n})})$
(71)

as $A\to\infty$, with $\bar{B}(A,B)=B/A^{\frac{2}{n}}=O(1)$. This completes
the structure to (IBVP) and (BVP) in the case of slowly varying fronts, with
$A\gg 1$ and $B\gg 1$.

We end this section with comments on the behaviour of the PTW propagation
speed. We observe from (66) and (71) that $\hat{v}(A,B,u_{c})$ is enhanced by
the shear flow through a prefactor that is entirely determined by the flow
whilst the effects of reaction cut-off are felt through the factor
$v^{*}(u_{c})$ corresponding to the propagation speed for the reaction cut-off
problem in the absence of a flow. Thus, the asymptotic limits (2) and (3) are,
in the presence of a shear flow, magnified by a factor entirely dependent on
the flow. For $B=O(A^{\frac{2}{n}})$, the enhancement is significant. Finally,
simple calculations for the Couette and Poiseulle flows give, specifically,

$\Delta=\begin{cases}1/120,&\quad\text{for}\quad\alpha(y)=y-1/2~{}(\text{Couette}),\\\
1/1890,&\quad\text{for}\quad\alpha(y)=-2y^{2}+2y-1/3~{}(\text{Poiseulle}).\end{cases}$
(72)

Thus, the PTW propagates faster in the plane Couette flow than in the plane
Poiseulle flow.

##  4 Slowly varying, balanced or rapidly varying front with weak advection:
$B^{-1}=o(1),B=O(1)$ or $B=o(1)$ as $A\to 0$

In this section we consider (IBVP) ((QIVP)) and (BVP ((QBVP)) in the case when
the advection velocity scale is small compared to advectionless front
propagation speed, so that $A\ll 1$. To begin with we examine the structure to
(IBVP) when also $B\gg 1$, so that the channel width is small compared to the
advectionless front thickness. To formalise this case, we consider (IBVP) when
$B^{-1}=o(1)$ as $A\to 0$.

###  4.1 Slowly varying front: $B^{-1}=o(1)$ as $A\to 0$

This limit considers the situation when the channel width is small compared to
the thickness of the advectionless front. For notational convenience, we write

$A=\delta$ (73)

and then a balance in (QIVPa) determines an expansion in the form,

$u(x,y,t;\delta)=u_{0}(x,y,t)+O(B^{-1}(\delta))$ (74)

as $\delta\to 0$ with $(x,y,t)\in\mathbb{R}\times[0,1]\times[0,\infty)$. Here,
formally, $B^{-1}(\delta)=o(1)$ as $\delta\to 0$. On substituting from (74)
into (IBVP), we obtain,

$u_{0}(x,y,t)=\bar{u}(x,t),~{}~{}(x,y,t)\in\mathbb{R}\times[0,1]\times[0,\infty),$
(75)

with $\bar{u}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ to be determined, with
suitable regularity. Moving to $O(B^{-1}(\delta))$, we proceed as in section
3, and, without repeating details, we obtain the following scalar problem for
$\bar{u}(x,t)$, namely,

$\bar{u}_{t}=\bar{u}_{xx}+f_{c}(\bar{u}),~{}~{}(x,t)\in\mathbb{R}\times(0,\infty),$
(76)

together with initial and boundary conditions (59) and (58). This is precisely
the scalar evolution problem studied in detail in [24, 25], where it is
established that $\bar{u}$ evolves into a PTW structure as $t\to\infty$, and,
for each $u_{c}\in(0,1)$, this PTW solution is unique, and has propagation
speed $v^{\ast}(u_{c})$, as given in [24]. Thus in this case, for (BVP),

$\hat{v}(A,B,u_{c})=v^{\ast}(u_{c})+O\left(A^{2}B^{-1}(A)\right)$ (77)

as $A\to 0$ with $B^{-1}(A)=o(1)$. This simple case is now complete. To
summarise this case, it follows from (75) that the deformation of the front
interface is weak and of order $O(AB^{-1}(A))$ whilst from (77) the correction
to the propagation speed is even weaker. This correction can be considered via
restricting attention to the problem (QBVP) since at higher order the detailed
nature of the discontinuity in the reaction must be addressed and so, the
free-boundary formulation (QBVP) needs to be adopted. We now move on to the
next case.

###  4.2 Balanced front: $B=O(1)$ as $A\to 0$

This limit addresses the situation when the channel width is comparable to the
front thickness in the absence of advection. In considering the limit $B=O(1)$
as $A\to 0$, since the discontinuity in the cut-off nonlinearity is
encountered immediately, at leading order, it is most convenient to address
the formulation (QIVP), and we restrict attention to PTW solutions to (QIVP),
via the formulation (QBVP). We look for a solution to (QBVP) expanded in the
form (with $A=\delta$),

$U_{T}(\xi^{\prime},y^{\prime};\delta)=u_{0}(\xi^{\prime},y^{\prime})+\delta
u_{1}(\xi^{\prime},y^{\prime})+O(\delta^{2})$ (78a) as $\delta\to 0$, with
$(\xi^{\prime},y^{\prime})\in Q_{L}\cap Q_{R}$. In addition, for
$y^{\prime}\in[0,L]$, we write,
$\zeta(y^{\prime},\delta)=\zeta_{0}(y^{\prime})+\delta\zeta_{1}(y^{\prime})+O(\delta^{2})$
(78b) and expand the propagation speed, $v(\delta)=v_{0}+\delta
v_{1}+O(\delta^{2}).$ (78c)

We now substitute from (78) into (QBVP). Collecting terms at $O(1)$, we obtain
the following problem for $u_{0}$, $\zeta_{0}$ and $v_{0}$, namely,

$\nabla^{2}u_{0}+v_{0}u_{0\xi^{\prime}}+f_{c}(u_{0})=0,~{}~{}(\xi^{\prime},y^{\prime})\in
Q_{L}\cup Q_{R}$ (79a) with $u_{0}\geq
u_{c}~{}\text{in}~{}Q_{L},~{}~{}u_{0}\leq u_{c}~{}\text{in}~{}Q_{R},$ (79b)
and
$u_{0}(\xi^{\prime},y^{\prime})\to\begin{cases}1~{}~{}\text{as}~{}~{}\xi^{\prime}\to-\infty,\\\
0~{}~{}\text{as}~{}~{}\xi^{\prime}\to\infty,\end{cases}$ (79c) uniformly for
$y^{\prime}\in[0,L]$, whilst
$u_{0y^{\prime}}(\xi^{\prime},0)=u_{0y^{\prime}}(\xi^{\prime},L)=0,~{}~{}\xi^{\prime}\in\mathbb{R}.$
(79d)
$u_{0}(\zeta_{0}(y^{\prime})^{+},y^{\prime})=u_{0}(\zeta_{0}(y^{\prime})^{-},y^{\prime})=u_{c},~{}~{}y^{\prime}\in[0,L],$
(79e)
$[(u_{0\xi^{\prime}}-\zeta_{0y^{\prime}}u_{0y^{\prime}})(\zeta_{0}(y^{\prime}),y^{\prime})]_{L}^{R}=0,~{}~{}y^{\prime}\in(0,L).$
(79f) $\int_{0}^{L}{\zeta_{0}(y^{\prime})}dy^{\prime}=0.$ (79g)

The elliptic problem (79) has a unique solution, and this solution is
independent of $y^{\prime}$. The solution is given in [24] (see Theorem 1) in
terms of $U_{T}:\mathbb{R}\to\mathbb{R}$, namely,

$u_{0}(\xi^{\prime},y^{\prime})=U_{T}(\xi^{\prime}),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L]$
(80a) with $\zeta_{0}(y^{\prime})=0~{}~{}\forall~{}~{}y^{\prime}\in[0,L]$
(80b) and $v_{0}=v^{\ast}(u_{c}).$ (80c)

We recall, from [24], that $U_{T}(\xi^{\prime})$ is monotone decreasing in
$\xi^{\prime}$, with,

$U_{T}^{\prime\prime}(0^{+})-U_{T}^{\prime\prime}(0^{-})=-f_{c},$ (81a)
$U_{T}(\xi^{\prime})=u_{c}e^{-v^{\ast}(u_{c})\xi^{\prime}}~{}~{}\forall~{}~{}\xi^{\prime}\in[0,\infty),$
(81b) $U_{T}(\xi^{\prime})\sim
1-A_{-\infty}e^{\lambda_{+}(v^{\ast}(u_{c}))\xi^{\prime}}~{}~{}\text{as}~{}~{}\xi^{\prime}\to-\infty,$
(81c) with the constant $A_{-\infty}>0$ and
$\lambda_{+}(v)=\frac{1}{2}\left((v^{2}+4|f_{c}^{\prime}(1)|)^{\frac{1}{2}}-v\right).$
(81d)

We now proceed to terms at $O(\delta)$ in (QBVP). This results in the
following inhomogeneous linear elliptic boundary value problem for $u_{1}$,
$\zeta_{1}$ and $v_{1}$, namely,

$\nabla^{2}u_{1}+v^{\ast}u_{1\xi^{\prime}}+H(-\xi^{\prime})f_{c}^{\prime}(U_{T}(\xi^{\prime}))u_{1}=(\bar{\alpha}(y^{\prime})-v_{1})U_{T}^{\prime}(\xi^{\prime}),~{}~{}(\xi^{\prime},y^{\prime})\in(\mathbb{R}\setminus\\{0\\})\times(0,L),$
(82a) $u_{1}(\xi^{\prime},y^{\prime})\to
0~{}~{}\text{as}~{}~{}|\xi^{\prime}|\to\infty~{}~{}\text{uniformly
for}~{}~{}y\in[0,L],$ (82b)
$u_{1y^{\prime}}(\xi^{\prime},0)=u_{1y^{\prime}}(\xi^{\prime},L)=0,~{}~{}\xi^{\prime}\in\mathbb{R},$
(82c)
$u_{1}(0^{+},y^{\prime})=u_{1}(0^{-},y^{\prime})=v^{\ast}u_{c}\zeta_{1}(y^{\prime}),~{}~{}y^{\prime}\in(0,L),$
(82d)
$u_{1\xi^{\prime}}(0^{+},y^{\prime})-u_{1\xi^{\prime}}(0^{-},y^{\prime})=-f_{c}\zeta_{1}(y^{\prime}),~{}~{}y^{\prime}\in(0,L),$
(82e) where (82d) and (82e) were obtained using (81a) and (81b).

We first eliminate $\zeta_{1}(y^{\prime})$ from the problem (82b)-(82e), which
reduces the boundary conditions (82d) and (82e) to

$u_{1}(0^{+},y^{\prime})=u_{1}(0^{-},y^{\prime}),$ (83a)
$u_{1\xi^{\prime}}(0^{+},y^{\prime})-u_{1\xi^{\prime}}(0^{-},y^{\prime})=-f_{c}(v^{\ast}u_{c})^{-1}u_{1}(0^{-},y^{\prime}),$
(83b)

for all $y^{\prime}\in(0,L)$, after which $\zeta_{1}(y^{\prime})$ is given by

$\zeta_{1}(y^{\prime})=(v^{\ast}u_{c})^{-1}u_{1}(0^{-},y^{\prime}),~{}~{}y^{\prime}\in[0,L].$
(84)

The linear elliptic problem (82) and (83) can be solved explicitly. We begin
by using Fourier’s Theorem to write,

$u_{1}(\xi^{\prime},y^{\prime})=\sum_{n=0}^{\infty}a_{n}(\xi^{\prime})\text{cos}\left(\frac{n\pi}{L}y^{\prime}\right),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L],$
(85)

with $a_{n}:\mathbb{R}\to\mathbb{R}$ ($n=0,1,2,...$) to be determined. We
observe that the boundary conditions (82c) are satisfied by (85). In addition,
we may write, via (QIVPj),

$\bar{\alpha}(y^{\prime})=\sum_{n=1}^{\infty}\bar{\alpha}_{n}\text{cos}\left(\frac{n\pi}{L}y^{\prime}\right),~{}~{}y^{\prime}\in[0,L],$
(86a) with
$\bar{\alpha}_{n}=\frac{2}{L}\int_{0}^{L}{\bar{\alpha}(s)\text{cos}\left(\frac{n\pi}{L}s\right)}ds$
(86b)

for $n=1,2,\ldots$. Now substitute from (85) and (86a) into (82a)–(82d), (83a)
and (83b). At $n=0$ we obtain the following problem for $a_{0}(\xi^{\prime})$,

$a_{0}^{\prime\prime}+v^{\ast}a_{0}^{\prime}+H(-\xi^{\prime})f^{\prime}(U_{T}(\xi^{\prime}))a_{0}=-v_{1}U_{T}^{\prime}(\xi^{\prime}),~{}~{}\xi^{\prime}\in\mathbb{R}\setminus\\{0\\},$
(87a) $a_{0}(0^{+})=a_{0}(0^{-}),$ (87b)
$a_{0}^{\prime}(0^{+})-a_{0}^{\prime}(0^{-})=-f_{c}(v^{\ast}u_{c})^{-1}a_{0}(0^{-}),$
(87c) $a_{0}(\xi^{\prime})\to 0~{}~{}\text{as}~{}~{}|\xi^{\prime}|\to\infty.$
(87d)

To solve (87), we first observe that $U_{T}^{\prime}(\xi^{\prime})$ is a
solution to the homogeneous form of the linear ODE (87a). After writing this
ODE in self-adjoint form, it is then readily established that the linear,
inhomogeneous, boundary value problem (87a)–(87d) has the _solvability
condition_

$v_{1}\left(\int_{-\infty}^{0}{(U_{T}^{\prime}(s))^{2}e^{v^{\ast}s}}ds+v^{\ast}u_{c}^{2}\right)=0$
(88)

which requires

$v_{1}=0.$ (89)

The solution to (87) is then given by,

$a_{0}(\xi^{\prime})=CU_{T}^{\prime}(\xi^{\prime}),~{}~{}\xi^{\prime}\in\mathbb{R},$
(90)

with $C$ a constant to be determined. Now, from (33) and (78b), we require

$\int_{0}^{L}{\zeta_{1}(y^{\prime})}dy^{\prime}=0,$ (91)

which becomes, via (84),

$\int_{0}^{L}{u_{1}(0^{-},y^{\prime})}dy^{\prime}=0$ (92)

and so, via (85) and (87b),

$a_{0}(0^{+})=a_{0}(0^{-})=0.$ (93)

Now $U_{T}^{\prime}(0)=-v^{\ast}u_{c}<0$, and so (90) and (93) requires $C=0$,
and therefore

$a_{0}(\xi^{\prime})=0,~{}~{}\xi^{\prime}\in\mathbb{R}.$ (94)

For $n=1,2,3...$, we obtain the problem,

$a_{n}^{\prime\prime}+v^{\ast}a_{n}^{\prime}-(n^{2}\pi^{2}L^{-2}-H(-\xi^{\prime})f^{\prime}(U_{T}(\xi^{\prime})))a_{n}=\bar{\alpha}_{n}U_{T}^{\prime}(\xi^{\prime}),~{}~{}\xi^{\prime}\in\mathbb{R}\setminus\\{0\\},$
(95a) $a_{n}(0^{+})=a_{n}(0^{-}),$ (95b)
$a_{n}^{\prime}(0^{+})-a_{n}^{\prime}(0^{-})=-f_{c}(v^{\ast}u_{c})^{-1}a_{n}(0^{-}),$
(95c) $a_{n}(\xi^{\prime})\to 0~{}~{}\text{as}~{}~{}|\xi^{\prime}|\to\infty.$
(95d)

The solution to (95) is readily obtained as,

$a_{n}(\xi^{\prime})=-\bar{\alpha}_{n}L^{2}(n^{2}\pi^{2})^{-1}U_{T}^{\prime}(\xi^{\prime}),~{}~{}\xi^{\prime}\in\mathbb{R}.$
(96)

Thus, via (94) and (96), we have, from (85),

$u_{1}(\xi^{\prime},y^{\prime})=-\frac{U_{T}^{\prime}(\xi^{\prime})L^{2}}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\bar{\alpha}_{n}}{n^{2}}\text{cos}\left(\frac{n\pi}{L}y^{\prime}\right),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L].$
(97)

We recall that $\bar{\alpha}(y^{\prime})$ is given by (86), and so, it is
convenient to introduce $\bar{a}(y^{\prime})$ as,

$\bar{a}(y^{\prime})\equiv\int_{0}^{y^{\prime}}{\bar{\alpha}(s)}ds\\\
=\frac{L}{\pi}\sum_{n=1}^{\infty}\frac{\bar{\alpha}_{n}}{n}\text{sin}\left(\frac{n\pi}{L}y^{\prime}\right),~{}~{}y^{\prime}\in[0,L].$
(98)

Integrating (98) and rearranging yields

$\frac{L^{2}}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\bar{\alpha}_{n}}{n^{2}}\text{cos}\left(\frac{n\pi}{L}y^{\prime}\right)=\frac{L^{2}}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\bar{\alpha}_{n}}{n^{2}}-\int_{0}^{y^{\prime}}{\bar{a}(s)}ds,~{}~{}y^{\prime}\in[0,L].$
(99)

Thus we have,

$u_{1}(\xi^{\prime},y^{\prime})=\left(\int_{0}^{y^{\prime}}{\bar{a}(s)}ds-\frac{L^{2}}{\pi^{2}}\sum_{n=1}^{\infty}{\frac{\bar{\alpha}_{n}}{n^{2}}}\right)U_{T}^{\prime}(\xi^{\prime}),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L].$
(100)

Now, from (99), we obtain,

$\frac{1}{L}\int_{0}^{L}\left(\int_{0}^{y^{\prime}}{\bar{a}(s)}ds\right)dy^{\prime}=\frac{L^{2}}{\pi^{2}}\sum_{n=1}^{\infty}{\frac{\bar{\alpha}_{n}}{n^{2}}}.$
(101)

It is thus convenient to introduce

$\phi(y^{\prime})=\int_{0}^{y^{\prime}}{\bar{a}(s)}ds=\int_{0}^{y^{\prime}}\left(\int_{0}^{w}{\bar{\alpha}(s)}ds\right)dw,~{}~{}y^{\prime}\in[0,L].$
(102)

We can now use (101) and (102) to re-write (100) as,

$u_{1}(\xi^{\prime},y^{\prime})=\left(\phi(y^{\prime})-\langle\phi\rangle_{L}\right)U_{T}^{\prime}(\xi^{\prime}),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L],$
(103)

with $\langle\cdot\rangle_{L}$ denoting the usual mean value on the interval
$[0,L]$. Finally, from (84) and (103), we obtain,

$\zeta_{1}(y^{\prime})=\langle\phi\rangle_{L}-\phi(y^{\prime}),~{}~{}y^{\prime}\in[0,L].$
(104)

Thus, in this case we have, via (78), (80), (89), (94), (103) and (104),

$U_{T}(\xi^{\prime},y^{\prime},\delta)=U_{T}(\xi^{\prime})+\delta\left(\phi(y^{\prime})-\langle\phi\rangle_{L}\right)U_{T}^{\prime}(\xi^{\prime})+O(\delta^{2}),~{}~{}(\xi^{\prime},y^{\prime})\in\mathbb{R}\times[0,L],$
(105a)
$\zeta(y^{\prime},\delta)=\delta(\langle\phi\rangle_{L}-\phi(y^{\prime})),~{}~{}y^{\prime}\in[0,L].$
(105b) $v(\delta)=v^{\ast}(u_{c})+O(\delta^{2}),$ (105c)

as $\delta\to 0$ with $L=B^{-\frac{1}{2}}=O(1)^{+}$. We conclude in this case
that there is a unique PTW solution, given by (105a) and (105b), for each
$u_{c}\in(0,1)$, which has propagation speed,

$\hat{v}(A,B,u_{c})=v^{\ast}(u_{c})+O(A^{2})$ (106)

as $A\to 0$ with $B=O(1)$. We note that, although the correction to the PTW
propagation speed is at least as small as $O(A^{2})$, the corrections to the
PTW structure are both at $O(A)$. We also note from (105b) with (102) and (98)
that $\zeta(y^{\prime},\delta)$ has interior stationary points if and only if
$\bar{a}(y^{\prime})$ has a zero in the interior of the domain. This completes
the asymptotic analysis in the case $A\to 0$ with $B=O(1)$. Contrasting with
the previous case, the correction to the propagation speed is of the same
order when $B(A)=O(1)$ as $A\to 0$, which is as expected. In fact we see that
in both cases the correction to the propagation speed can be written,
uniformly as $O(A^{2}/(B(A)+A))$ as $A\to 0$, and we will see that this
continues to hold in each of the cases considered below in the rest of this
section. We finally compute the interface deformation for the Couette flow
(8a) and the Poiseuille flow (8b) using (105b) and (102). Figure 2 compares
the deformation against the structure of the two flows. It is clear that the
deformation in the Couette flow is more pronounced than in the plane Poiseulle
flow.

Figure 2: A graph of the interface $\zeta(y)/A$ as $A\to 0$ obtained for $B=1$
using (105b) and (102) for the Couette (thick solid lines) and the Poiseuille
flow (thin solid lines).

The final case in this section is $A\ll 1$ with $B\ll 1$, and the appropriate
distinguished limit is $B=O(A)$ as $A\to 0$.

###  4.3 Rapidly varying front: $B=O(A)$ as $A\to 0$

Here we consider the asymptotic structure of PTW solutions when the front
thickness in the absence of advection is small compared to the channel width.
In this case, it is natural to consider PTW solutions via the formulation
(BVP). Again, setting $A=\delta$, we write,

$B=\bar{B}\delta$ (107)

with $\bar{B}=O(1)$ as $\delta\to 0$. We now expand in the form,

$U_{T}(z,y;\delta)=u_{0}(z,y)+\delta
u_{1}(z,y)+O(\delta^{2}),~{}~{}(z,y)\in\mathbb{R}\times[0,1],$ (108a)
$\gamma(y;\delta)=\gamma_{0}(y)+\delta\gamma_{1}(y)+O(\delta^{2}),~{}~{}y\in[0,1],$
(108b) $v(\delta)=v_{0}+\delta v_{1}+O(\delta^{2})$ (108c)

as $\delta\to 0$. We substitute (107) and (108) into (BVP). At leading order
we obtain the following problem for $u_{0}$, $v_{0}$ and $\gamma_{0}$, namely,

$u_{0zz}+v_{0}u_{0z}+f_{c}(u_{0})=0,~{}~{}(z,y)\in\mathbb{R}\times(0,1),$
(109a) $u_{0}(\gamma_{0}^{\pm}(y),y)=u_{c},~{}~{}y\in[0,1],$ (109b)
$u_{0z}(\gamma_{0}^{+}(y),y)=u_{0z}(\gamma_{0}^{-}(y),y),~{}~{}y\in[0,1],$
(109c) $u_{0}(z,y)\to\begin{cases}1~{}~{}\text{as}~{}~{}z\to-\infty,\\\
0~{}~{}\text{as}~{}~{}z\to\infty,\end{cases}$ (109d) uniformly for
$y\in[0,1]$, $u_{0y}(z,0)=u_{0y}(z,1)=0,~{}~{}z\in\mathbb{R},$ (109e)
$\int_{0}^{1}{\gamma_{0}(y)}dy=0.$ (109f)

Since partial derivatives with respect to $y$ do not appear in the problem
(109a)–(109d) we can address this problem following [24] and regard $y$ as a
parameter. Thus, we may conclude that, up to translation invariance, (109) has
a unique solution. Since we may regard the translational invariance as
dependent on $y$ then we may write the solution as

$u_{0}(z,y)=U_{T}(z-\gamma_{0}(y)),~{}~{}(z,y)\in\mathbb{R}\times[0,1],$ (110)
$v_{0}=v^{\ast}(u_{c}),$ (111)

with $\gamma_{0}:[0,1]\to\mathbb{R}$ to be determined, so that $\gamma_{0}\in
C^{1}([0,1])$, and satisfies condition (109f), together with,

$\gamma_{0}^{\prime}(0)=\gamma_{0}^{\prime}(1)=0$ (112)

via (110) with (109e). We next formulate the problem at $O(\delta)$ from
(BVP), which provides an inhomogeneous linear problem for $u_{1}$ and
$\gamma_{1}$. After a significant amount of detailed, but routine, calculation
on this problem (which, for brevity, we do not include here), we find that it
requires a solvability condition arising from the classical Fredholm
alternative which must be satisfied for the linear inhomogenous boundary value
problem to have a solution. This provides a nonlinear eigenvalue problem which
determines $\gamma_{0}:[0,1]\to\mathbb{R}$ and $v_{1}\in\mathbb{R}$, namely,

$\gamma_{0}^{\prime\prime}+\frac{1}{2}v^{\ast}(\gamma_{0}^{\prime})^{2}+\bar{B}^{-1}(\alpha(y)-v_{1})=0,~{}~{}y\in(0,1),$
(113)

together with boundary conditions (112) and condition (109f), which we
henceforth refer to as (EP). In considering (EP), we first introduce
$\psi:[0,1]\to\mathbb{R}$ given by

$\psi(y)=\text{exp}\left(\frac{1}{2}v^{\ast}\gamma_{0}(y)\right)~{}~{}\forall~{}~{}y\in[0,1].$
(114)

which requires that

$\psi(y)>0,~{}~{}y\in[0,1]$ (115)

and, upon using (109f), that

$\int_{0}^{1}{\log(\psi(y))}dy=0.$ (116)

In terms of this new dependent variable, (EP) becomes a classical regular
Sturm-Liouville eigenvalue problem (see, for example, Coddington and Levinson
[5]) that we will henceforth refer to as (SL):

$\psi^{\prime\prime}+(k\alpha(y)+\lambda)\psi=0,~{}~{}y\in(0,1),$ (SLa) with
boundary conditions $\psi^{\prime}(0)=\psi^{\prime}(1)=0,$ (SLb) and condition
where
$\lambda=-\frac{1}{2}v^{\ast}\bar{B}^{-1}v_{1},~{}~{}k=\frac{1}{2}v^{\ast}\bar{B}^{-1}.$

Indeed, (SL) has arisen via a very different route in the work of Haynes and
Vanneste [14] in the context of a purely advection-diffusion problem at high
Péclet number (corresponding to high $A/\sqrt{B}$), and in relation to the
present context, it has been studied by them for the particular Couette and
plane Poiseuille flows noted in (8). The positivity requirement (115) dictates
that we require the _smallest (principal)_ eigenvalue of this Sturm-Liouville
problem. We denote the principal eigenvalue by

$\lambda=\lambda_{0}(k)$ (117)

and the associated principal, $L^{1}$-normalised, eigenfunction by
$\psi=\psi_{0}:[0,1]\to\mathbb{R}$ with

$\psi_{0}=\psi_{0}(y,k)>0~{}~{}\forall~{}~{}y\in[0,1]$ (118)

and

$\int_{0}^{1}{\psi_{0}(y,k)}dy=1.$ (119)

with the choice of $L^{1}$ normalisation being convenient at a later stage.
The solution to (EP) is then given by, on satisfying the final condition
(116),

$\gamma_{0}(y)=\frac{2}{v^{\ast}}\left(\log(\psi_{0}(y,k))-\langle\log(\psi_{0}(\cdot,k))\rangle\right)~{}~{}\forall~{}~{}y\in[0,1],$
(120)

with

$v_{1}=-\frac{\lambda_{0}(k)}{k}.$ (121)

We now consider (SL) in the two cases when $k\gg 1$ and $k\ll 1$.

####  4.3.1 (SL) when $k\ll 1$

We consider (SL) as $k\to 0$. An exposition of this type has been developed in
[14], and for completeness we give a brief development in the present context.
It follows from (SL) with (118) and (119) that

$\lambda_{0}(k)\to 0,\quad\text{and}\quad\psi_{0}(y,k)\to
1~{}~{}\text{uniformly for}~{}~{}y\in[0,1],$ (122)

as $k\to 0$. Thus we expand in the form,

$\psi_{0}(y,k)=1+k\bar{\psi}_{0}(y)+O(k^{2}),~{}~{}y\in[0,1],$ (123a)
$\lambda_{0}(k)=k\bar{\lambda}_{0}+O(k^{2}),$ (123b)

as $k\to 0$. On substitution into (SL) using (119), we obtain the problem

$\bar{\psi}_{0}^{\prime\prime}+(\alpha(y)+\bar{\lambda}_{0})=0,~{}~{}y\in(0,1),$
(124a) $\bar{\psi}_{0}^{\prime}(0)=\bar{\psi}_{0}^{\prime}(1)=0,$ (124b)
$\int_{0}^{1}{\bar{\psi}_{0}(y)}dy=0.$ (124c)

The solution to (124) is readily obtained as

$\bar{\psi}_{0}(y)=\left(\langle\widetilde{\alpha}(\cdot)\rangle-~{}\widetilde{\alpha}(y)\right)~{}~{}\forall~{}~{}y\in[0,1],$
(125a) with $\widetilde{\alpha}:[0,1]\to\mathbb{R}$ given by (62), and,
$\bar{\lambda}_{0}=0.$ (125b)

Thus, via (120), (121), (123) and (125), we obtain,

$v_{1}=O(k),$ (126)
$\gamma_{0}(y)=\frac{2k}{v^{\ast}(u_{c})}\left(\langle\widetilde{\alpha}(\cdot)\rangle-~{}\widetilde{\alpha}(y)\right)+O(k^{2})~{}~{}\forall~{}~{}y\in[0,1],$
(127)

as $k\to 0$. It should be noted that these results, _for $\bar{B}\gg 1$_, are
in full accord with those of section 4.2, _for $B=O(1)$_. Expression (126) can
be made more precise by performing higher-order corrections to
$\lambda_{0}(k)$, the details of which are presented in [14] and give

$\lambda_{0}(k)=-k^{2}\left\langle\left(\int_{0}^{y}\alpha(s)ds\right)^{2}\right\rangle+O(k^{3}),$
(128)

as $k\to 0$. Upon using (121) these corrections yield

$v_{1}=k\left\langle\left(\int_{0}^{y}\alpha(s)ds\right)^{2}\right\rangle+O(k^{2}),$
(129)

as $k\to 0$.

####  4.3.2 (SL) when $k\gg 1$

We consider (SL) as $k\to\infty$. An analysis of this type has been given for
the Couette and plane Poiseuille flow in [14]. Here we develop the theory to
apply to any shear flow which has a continuous derivative on $[0,1]$, and
whose _absolute maximum on $[0,1]$ occurs strictly at an interior point_.
First, let

$\alpha_{M}=\sup_{y\in[0,1]}\alpha(y)>0$ (130)

when $\alpha(y)$ is nontrivial, via (IBVPd). Now, we restrict attention to the
situation when $\alpha_{M}$ is achieved ($\alpha\in C^{1}([0,1])$) at a
_single point, which is in the interior of $[0,1]$_. We label this point as
$y_{M}\in(0,1)$. Numerical experiments on (SL) with $k\gg 1$ suggest that the
principal eigenfunction, $\psi_{0}$, under the normalisation (119), develops
into a $\delta$-sequence as $k\to\infty$, based at $y=y_{M}$. This leads us to
write

$\psi_{0}(y,k)\sim\delta(y-y_{M}),~{}~{}y\in[0,1]$ (131)

as $k\to\infty$, which automatically satisfies the normalisation (119). Now,
upon integration and using (119), (SL) becomes

$k\int_{0}^{1}{\alpha(y)\psi_{0}(y,k)}dy+\lambda_{0}(k)=0$

which gives, using (131), $\lambda_{0}(k)\sim-\alpha_{M}k$ as $k\to\infty$.
This suggests the principal eigenvalue of (SL) should be expanded as,

$\lambda_{0}(k)=-\alpha_{M}k+\tilde{\lambda}k^{\frac{1}{2}}+O(1)$ (132)

as $k\to\infty$, with $\widetilde{\lambda}\in\mathbb{R}$ to be determined.
Using this, we may now determine the structure of
$\psi_{0}:[0,1]\to\mathbb{R}$ as $k\to\infty$, in more detail, together with
determining $\widetilde{\lambda}$. We first anticipate, from (SL), that there
will develop boundary layers at $y=0$ and $y=1$, as $k\to\infty$. We focus
attention first at $y=0$. The boundary layer thickness will be
$y=O(k^{-\frac{1}{2}})$, and so we introduce

$Y=k^{\frac{1}{2}}y=O(1)^{+}$ (133)

as $k\to\infty$ in the boundary layer. At leading order, (SL) becomes

$\psi_{0YY}-(\alpha_{M}-\alpha(0))\psi_{0}=0,~{}~{}Y>0,$ (134a)
$\psi_{0Y}(0)=0.$ (134b)

The solution to (134) has,

$\psi_{0}(Y,k)\sim
A^{L}(k)\cosh\left(\left(\alpha_{M}-\alpha(0)\right)^{\frac{1}{2}}Y\right)$
(135)

as $k\to\infty$ with $Y=O(1)^{+}$, and $A^{L}(k)$ a normalising factor to be
determined. Similarly, for the boundary layer at $y=1$, we introduce,

$\bar{Y}=(y-1)k^{\frac{1}{2}}=O(1)^{-}$ (136)

as $k\to\infty$ in the boundary layer, after which we obtain,

$\psi_{0}(\bar{Y},k)\sim
A^{R}(k)\cosh\left(\left(\alpha_{M}-\alpha(0)\right)^{\frac{1}{2}}\bar{Y}\right)$
(137)

as $k\to\infty$ with $\bar{Y}=O(1)^{-}$, and $A^{R}(k)$ a normalising factor
to be determined. The remaining asymptotic structure now has three regions, as
follows:

  * •

region $L$ :
$y\in\left(O(k^{-\frac{1}{2}})^{+},y_{M}-O(k^{-\frac{1}{4}})^{+}\right)$

  * •

region $R$:
$y\in\left(y_{M}+O(k^{-\frac{1}{4}})^{+},1-O(k^{-\frac{1}{2}})^{+}\right)$

  * •

spike region :
$y\in\left(y_{M}-O(k^{-\frac{1}{4}})^{+},y_{M}+O(k^{-\frac{1}{4}})^{+}\right)$.

We first move to region $L$. The form of solution (135) in the boundary layer
at $y=0$, leads to a WKB form for the solution in this region, which, after
asymptotic matching with (135), gives,

$\psi_{0}(y,k)\sim
A^{L}(k)\text{cosh}\left(k^{\frac{1}{2}}\int_{0}^{y}{(\alpha_{M}-\alpha(s))^{\frac{1}{2}}}ds\right)$
(138)

as $k\to\infty$ with
$y\in\left(O(k^{-\frac{1}{2}})^{+},y_{M}-O(k^{-\frac{1}{4}})^{+}\right)$.
Similarly, in region $R$ we have,

$\psi_{0}(y,k)\sim
A^{R}(k)\text{cosh}\left(k^{\frac{1}{2}}\int_{y}^{1}{(\alpha_{M}-\alpha(s))^{\frac{1}{2}}}ds\right)$
(139)

as $k\to\infty$ with
$y\in\left(y_{M}+O(k^{-\frac{1}{4}})^{+},1-O(k^{-\frac{1}{2}})^{+}\right)$. We
finally move to the spike region. In this region we introduce

$w=(y-y_{M})k^{\frac{1}{4}}=O(1)$ (140)

as $k\to\infty$. The expansions (138) and (139) require us to write

$\psi_{0}(w,k)=A(k)\widetilde{\psi}_{0}(w)+o(A(k))$ (141)

as $k\to\infty$ with $w=O(1)$. Here $A(k)\to\infty$ as $k\to\infty$ is to be
determined. Substitution from (140), (141) and (132) into equation (SLa)
gives, at leading order,

$\widetilde{\psi}_{0ww}-\left(\frac{1}{2}(-\alpha_{M}^{\prime\prime})w^{2}-\widetilde{\lambda}\right)\widetilde{\psi}_{0}=0,~{}~{}w\in\mathbb{R},$
(142a) $\widetilde{\psi}_{0}(w)>0~{}~{}\forall~{}~{}w\in\mathbb{R},$ (142b)
whilst matching with region $L$ and region $R$ (via Van Dyke’s matching
principle [27]) requires, on using (138) and (139),
$\widetilde{\psi}_{0}(w)\sim\text{exp}\left(-\frac{(-\alpha_{M}^{\prime\prime})^{\frac{1}{2}}}{2^{\frac{3}{2}}}w^{2}\right)~{}~{}\text{as}~{}~{}|w|\to\infty,$
(142c)

and also that $A^{L}(k)$, $A^{R}(k)$ and $A(k)$ satisfy the two conditions,

$A^{L}(k)\text{exp}\left(k^{\frac{1}{2}}\int_{0}^{y_{M}}{(\alpha_{M}-\alpha(s))^{\frac{1}{2}}}ds\right)=A(k),$
(143)
$A^{R}(k)\text{exp}\left(k^{\frac{1}{2}}\int_{y_{M}}^{1}{(\alpha_{M}-\alpha(s))^{\frac{1}{2}}}ds\right)=A(k).$
(144)

In addition the $L^{1}([0,1])$ normalising condition (119) requires,

$A(k)\int_{-\infty}^{\infty}{\widetilde{\psi}_{0}(w)}dw=k^{\frac{1}{4}}.$
(145)

In the above

$\alpha_{M}^{\prime\prime}=\alpha^{\prime\prime}(y_{M})<0$ (146)

restricting attention to a nondegenerate interior maximum for $\alpha(y)$ (the
case when the interior maximum is degenerate can be similarly addressed). Now,
(142a)-(142c) is a linear self-adjoint singular Sturm-Liouville eigenvalue
problem on the whole real line. The positivity condition (142b) determines
that _$\widetilde{\lambda}$ must be the smallest (principal) eigenvalue_. The
condition (145) then normalises this eigenfunction. In fact, we can determine
the lowest eigenvalue and the associated eigenfunction by inspection, which
gives,

$\widetilde{\psi}_{0}(w)=\text{exp}\left(-\frac{(-\alpha_{M}^{\prime\prime})^{\frac{1}{2}}}{2^{\frac{3}{2}}}w^{2}\right)~{}~{}\forall~{}~{}w\in\mathbb{R},$
(147a) with,
$\widetilde{\lambda}=\frac{1}{\sqrt{2}}(-\alpha_{M}^{\prime\prime})^{\frac{1}{2}}.$
(147b)

It then follows, via (143)–(147a), that,

$A(k)=\frac{(-\alpha_{M}^{\prime\prime})^{\frac{1}{4}}}{2^{\frac{3}{4}}\sqrt{\pi}}k^{\frac{1}{4}},$
(148a)
$A^{L}(k)=\frac{(-\alpha_{M}^{\prime\prime})^{\frac{1}{4}}}{2^{\frac{3}{4}}\sqrt{\pi}}k^{\frac{1}{4}}\text{exp}\left(-k^{\frac{1}{2}}\int_{0}^{y_{M}}{(\alpha_{M}-\alpha(s))^{\frac{1}{2}}}ds\right),$
(148b)
$A^{R}(k)=\frac{(-\alpha_{M}^{\prime\prime})^{\frac{1}{4}}}{2^{\frac{3}{4}}\sqrt{\pi}}k^{\frac{1}{4}}\text{exp}\left(-k^{\frac{1}{2}}\int_{y_{M}}^{1}{(\alpha_{M}-\alpha(s))^{\frac{1}{2}}}ds\right).$
(148c)

Finally, via (132) and (147b), we have,

$\lambda_{0}(k)=-\alpha_{M}k+\frac{1}{\sqrt{2}}(-\alpha_{M}^{\prime\prime})^{\frac{1}{2}}k^{\frac{1}{2}}+O(1)$
(149)

as $k\to\infty$, and so (121) gives,

$v_{1}=\alpha_{M}-\frac{1}{\sqrt{2}}(-\alpha_{M}^{\prime\prime})^{\frac{1}{2}}k^{-\frac{1}{2}}+O(k^{-1})$
(150)

as $k\to\infty$. The analysis in this subcase is now complete. However, it is
instructive to note that the structure will be adjusted when $y_{M}\to 0$ or
$1$ as $k\to\infty$, and in particular, when $y_{M}=O(k^{-\frac{1}{4}})^{+}$
or $y_{M}=1-O(k^{-\frac{1}{4}})^{+}$ as $k\to\infty$. A detailed consideration
of the case when $y_{M}=O(k^{-\frac{1}{4}})^{+}$ as $k\to\infty$, reveals that
the leading order form of $\lambda_{0}(k)$ remains unchanged, but the
correction is influenced, so that, in these cases,

$\lambda_{0}(k)=-\alpha_{M}k+o(k)$ (151)

as $k\to\infty$. Evidently, the case of Couette flow falls into this category,
and the details for this case are developed in [14], where it is established
that the above correction is, in fact, of $O(k^{\frac{2}{3}})$. For brevity we
do not pursue these special cases further.

####  4.3.3 (SL) when $k=O(1)^{+}$

In this case, details of (SL) must be obtained numerically. However, since we
are dealing with the principal eigenvalue, whose normalised eigenfunction is
strictly positive, a simple integration of equation (SLa) over the interval
$[0,1]$, using conditions (SLb), and rearranging, establishes the following
bounds on $\lambda_{0}(k)$, namely

$-k\alpha_{M}<\lambda_{0}(k)<-k\alpha_{m}$ (152)

for all $k>0$, with

$\alpha_{m}=\inf_{y\in[0,1]}\alpha(y)<0,$ (153)

when $\alpha(y)$ is nontrivial. We can thus conclude, from (121), that,

$\alpha_{m}<v_{1}(k)<\alpha_{M}$ (154)

for all $k>0$. In fact, we can improve the upper bound on $\lambda_{0}(k)$, by
again using the strict positivity of the principal eigenfunction to establish,
after dividing through equation (SLa) by $\psi$ and, using integration by
parts together with (IBVPd), that

$\lambda_{0}(k)=-\int_{0}^{1}{\frac{(\psi_{0}^{\prime}(y,k))^{2}}{(\psi_{0}(y,k))^{2}}}dy<0$
(155)

for each $k>0$, when $\alpha(y)$ is nontrivial, and so we then have the
improved lower bound

$v_{1}(k)>0$ (156)

for all $k>0$.

Figure 3: A graph of the scaled leading-order correction to the speed as $A\to
0$ with $B=O(A)$ obtained from (SL) as a function of
$k^{-1}=(\frac{1}{2}v^{\ast}AB^{-1})^{-1}$ for the Couette flow (thick solid
lines) and the Poiseuille flow (thin solid lines).

It is instructive at this stage to summarise the asymptotic results concerning
PTW solutions as $A\to 0$, the case of weak advection. We have considered the
situation when $0<B\ll 1$, $B=O(1)$ and $B\gg 1$, in the natural distinguished
limits, as $A\to 0$. We have shown that, when $A\ll 1$, then for each $B>0$,
there exists a unique PTW solution to (BVP) (correspondingly (QBVP)), with
propagation speed denoted by $v=\hat{v}(A,B,u_{c})$. With regards to this
propagation speed, we have determined that, _as $A\to 0$_,

  * (i)

$B\gg 1$ or $B=O(1)$

$\hat{v}(A,B,u_{c})=v^{\ast}(u_{c})+O\left(\frac{A^{2}}{B}\right)$ (157)

  * (ii)

$B\ll 1$ ($B=O(A)$)

$\hat{v}(A,B,u_{c})=v^{\ast}(u_{c})-Ak^{-1}\left(\frac{A}{B}\right)\lambda_{0}\left(k\left(\frac{A}{B}\right)\right)+O(A^{2}).$
(158)

Here

$k(\mu)=\frac{1}{2}v^{\ast}(u_{c})\mu~{}~{}\forall~{}~{}\mu\in(0,\infty)$
(159)

and $\lambda_{0}(k)$ is the principal eigenvalue of the Sturm-Liouville
problem (SL). We have established, upon use of results in [14], that when
$k\to 0$ (so that $0<A\ll B\ll 1$), then,

$\lambda_{0}(k)=-k^{2}\left\langle\left(\int_{0}^{y}\alpha(s)ds\right)^{2}\right\rangle+O(k^{3})$
(160)

whilst when $k\to\infty$ (so that $0<B\ll A\ll 1$) then,

$k^{-1}\lambda_{0}(k)=-\alpha_{M}+\frac{1}{\sqrt{2}}(-\alpha_{M}^{\prime\prime})^{\frac{1}{2}}k^{-\frac{1}{2}}+O(k^{-1}).$
(161)

We observe that the effects of weak advection on the PTW propagation speed is
always of $o(1)$ as $A\to 0$, but becomes more significant with decreasing
$B$. It is also worth noting here that expansion (158) forms a composite
expansion for $\hat{v}(A,B,u_{c})$ as $A\to 0$ for all $B>0$, with error
uniformly of $O(A^{2})$. In addition, it follows from this observation and the
limiting forms (160) and (161), that the correction to the propagation speed
is of $O(A)$ when $0<B\leq O(A)$, initially decreasing from the value
$A\alpha_{M}$ as $A^{1/2}B^{1/2}$, and continuing to decrease with increasing
$B$, becoming of $O(A^{2})$ for $B\geq O(1)$, and decreasing at a rate of
$A^{2}B^{-1}$. We finally observe that the two complimentary asymptotic forms
for the propagation speed as $A\to 0$, given in (157) and (158) match with
each other according to Van Dyke’s asymptotic matching principle [27]. To
conclude this section, we now compare the moving boundary location in the PTW
solutions. Assembling the results of this section we obtain, _as $A\to 0$_,
that the moving boundary is at spatial location $z=\zeta(y,A,B)$ for
$y\in[0,1]$, with,

  * (i)

$B\gg 1$

$\zeta(y,A,B)=O\left(\frac{A}{B}\right),~{}~{}y\in[0,1]$ (162)

  * (ii)

$B=O(1)$

$\zeta(y,A,B)=\frac{1}{B}\left(\langle\phi(\cdot)\rangle-\phi(y)\right)A+O(A^{2}),~{}~{}y\in[0,1]$
(163a) with $\phi(y)=\int_{0}^{y}\left({\int_{0}^{s}{\alpha(p)}dp}\right)ds.$
(163b)

  * (iii)

$B\ll 1$ ($B=O(A)$)

$\zeta(y,A,B)=\frac{2}{v^{\ast}(u_{c})}(\log\psi_{0}(y,k)-\langle\log\psi_{0}(\cdot,k)\rangle)+O(A),~{}~{}y\in[0,1],$
(164)

with $k=\frac{1}{2}v^{\ast}(u_{c})AB^{-1}$. Here $\psi_{0}:[0,1]\to\mathbb{R}$
is the principal, $L^{1}$-normalised eigenfunction of (SL). The structure of
$\psi_{0}$ as $k\to 0$ ($A\ll B\ll 1$) and as $k\to\infty$ ($B\ll A\ll 1$) is
as given in subsections 4.3.1 and 4.3.2.

Figure 4: A graph of the scaled interface $2\zeta/v^{*}$ as $A\to 0$ with
$B=O(A)$ for the Couette flow (left) and the Poiseuille flow (right) obtained
by solving (SL) for $k=1/5$, $k=1$ and $k=5$. The maximum of the interface
decreases with increasing $k$.

Figure 3 shows the correction to speed of propagation of the PTW solution
(divided by $A$) computed by numerical solution of the (SL) eigenvalue problem
for the Couette and Poiseuille (8) flows. Clearly, the enhancement to the
speed is, for fixed (small) $A$, greater for the Couette flow than for the
Poiseuille flow, decreasing monotonically as either the front thickness or
cut-off value increase. Figure 4 shows the leading order scaled interface
$2\zeta(y,A,B)/v^{*}(u_{c})$, deduced from (120) for three different values of
$k=\frac{1}{2}v^{\ast}(u_{c})AB^{-1}$. The interface becomes increasingly flat
as either the front thickness or cut-off value increase.

We now move on to consider PTW solutions when advection and streamwise
diffusion are balanced.

##  5 Slowly varying or rapidly varying front with balanced advection:
$B\to\infty$ or $B\to 0$ with $A=O(1)$

In this section we consider (IBVP) ((QIVP)) and (BVP) ((QBVP)) in the final
case when the advective fluid velocity is comparable to the advectionless
front propagation speed, so that $A=O(1)$. We consider this situation first
when the channel width is small compared to the advectionless front thickness
($B\gg 1$), and then when the channel width is large compared to the
advectionless front thickness ($B\ll 1$). We begin with the former case.

###  5.1 Slowly varying front: $B\to\infty$

Following details very similar to those presented in section 3 (and without
repetition) we have, for (IBVP),

$u(x,y,t;A,B)=\bar{u}(x,t)+O(B^{-1})$ (165)

as $B\to\infty$, with $(x,y,t)\in\mathbb{R}\times[0,1]\times[0,\infty)$, where
$\bar{u}$ satisfies the one-dimensional evolution problem comprising the PDE
(64a), _with $\bar{B}^{-1}=0$_, subject to the initial and boundary conditions
(58) and (59). It then follows, regarding PTW solutions, that the propagation
speed has

$\hat{v}(A,B,u_{c})=v^{\ast}(u_{c})+O(B^{-1})$ (166)

as $B\to\infty$ with $A=O(1)$. With the PTW moving boundary at
$z=\zeta(y,A,B)$, it also follows from (165) that,

$\zeta(y,A,B)=O(B^{-1}),~{}~{}y\in[0,1]$ (167)

as $B\to\infty$ with $A=O(1)$.

###  5.2 Rapidly varying front: $B\to 0$

We restrict attention to PTW solutions, and the most convenient formulation to
work with is that given in (BVP). It is convenient to write

$B=\widetilde{\epsilon}^{2}.$ (168)

The form of the PDE (BVPa) with $A=O(1)$ as $\widetilde{\epsilon}\to 0$, leads
us to write,

$U_{T}(z,y;\widetilde{\epsilon})=u_{0}\left(\left(z-\frac{z_{0}(y)}{\widetilde{\epsilon}}\right)\psi(y)\right)+O(\widetilde{\epsilon})$
(169)

as $\widetilde{\epsilon}\to 0$ with $(z,y)\in\mathbb{R}\times[0,1]$, and
$z_{0},\psi:[0,1]\to\mathbb{R}$, $u_{0}:\mathbb{R}\to\mathbb{R}$ each having
$z_{0}$, $\psi$, $u_{0}$ $=O(1)$ as $\widetilde{\epsilon}\to 0$. In addition
we require,

$\psi(y)>0~{}~{}\forall~{}~{}y\in[0,1],$ (170)

via (BVPe). In addition to (169) we also expand,

$v(\widetilde{\epsilon})=v_{0}+O(\widetilde{\epsilon})$ (171)

as $\widetilde{\epsilon})$. On substitution from (169) and (171) into (BVP) we
obtain, at leading order,

$\begin{split}(\psi^{2}(y)(1+z_{0}^{\prime}(y)^{2})+&2z_{0}(y)z_{0}^{\prime}(y)\psi(y)\psi^{\prime}(y)+z_{0}^{2}(y)\psi^{\prime}(y)^{2})u_{0}^{\prime\prime}(\xi)\\\
&+\psi(y)(v_{0}-A\alpha(y))u_{0}^{\prime}(\xi)+f_{c}(u_{0}(\xi))=0,~{}~{}(\xi,y)\in\mathbb{R}\times(0,1),\end{split}$
(172)

with ′ representing differentiation in the respective arguement, and where
$\xi$, replacing $z$, is given by,

$\xi=\left(z-\frac{z_{0}(y)}{\widetilde{\epsilon}}\right)\psi(y).$ (173)

Since $\xi$ and $y$ are independent coordinates, the solubility of (172)
requires, without loss of generality,

$\psi^{2}(y)(1+z_{0}^{\prime}(y)^{2})+2z_{0}(y)z_{0}^{\prime}(y)\psi(y)\psi^{\prime}(y)+z_{0}^{2}(y)\psi^{\prime}(y)^{2}\\\
=1,$ (174a) $\psi(y)(v_{0}-A\alpha(y))=v^{\ast}(u_{c}),$ (174b) for
$y\in(0,1)$, on normalising $\psi$ so that, $\sup_{y\in[0,1]}\psi(y)=1.$
(174c) The boundary conditions in (BVP) require,
$\psi^{\prime}(0)=\psi^{\prime}(1)=z_{0}^{\prime}(0)=z_{0}^{\prime}(1)=0.$
(174d)

Before considering (174) in more detail, we first write down the problem for
$u_{0}(\xi)$, which becomes, via (174), and (170),

$u_{0}^{\prime\prime}(\xi)+v^{\ast}(u_{c})u_{0}^{\prime}(\xi)+f_{c}(u_{0}(\xi))=0,~{}~{}\xi\in\mathbb{R},$
(175a) subject to $u_{0}(\xi)\geq 0~{}~{}\forall~{}~{}\xi\in\mathbb{R},$
(175b) $u_{0}(\xi)\to\begin{cases}0~{}~{}\text{as}~{}~{}\xi\to\infty,\\\
1~{}~{}\text{as}~{}~{}\xi\to{-\infty}.\end{cases}$ (175c)

This problem has the solution (as noted in previous sections, and see also
[24]),

$u_{0}(\xi(z,y))=U_{T}(\xi(z,y)),~{}~{}(z,y)\in\mathbb{R}\times[0,1],$ (176)

and we recall from [24] that,

$U_{T}^{\prime}(s)<0~{}~{}\forall~{}~{}s\in\mathbb{R}.$ (177)

Now, following (169) and (176), the moving boundary, where $U_{T}=u_{c}$,
occurs at $\xi(z,y)=0$, and so, via (171), when

$z=\frac{z_{0}(y)}{\widetilde{\epsilon}},~{}~{}y\in[0,1]$ (178)

and so, we must have, via (BVP) ((30)),

$\int_{0}^{1}{z_{0}(y)}dy=0.$ (179)

To proceed we will limit attention to shear flows $\alpha(y)$ which have a
single point of maximum velocity $\alpha_{M}$, occuring at $y=y_{M}$, where,
without loss of generality, we may take $y_{M}\in[\frac{1}{2},1]$. Next we
observe from (174b) that the continuous functions $\psi(y)$ and $\alpha(y)$
achieve their maximum values on $[0,1]$, of unity and $\alpha_{M}$
respectively, at the same point $y=y_{M}$, after which (174b) evaluated at
$y=y_{M}$, requires,

$v_{0}=v^{\ast}(u_{c})+A\alpha_{M}.$ (180)

Equation (174b) then becomes

$\psi(y)=v^{\ast}(u_{c})(v^{\ast}(u_{c})+A(\alpha_{M}-\alpha(y)))^{-1}~{}~{}\forall~{}~{}y\in[0,1].$
(181)

Figure 5: A graph of the scaled interface $z\sqrt{B}$ as $B\to 0$ for $A=1$
for the (left) Couette flow (8a) and (right) Poiseuille flow (8b). These are
numerically obtained using (185) and (179) for $v^{*}(u_{c})=0.1$, $1$ and
$1.9$. The maximum of the interface decreases with increasing $v^{*}(u_{c})$.

We are left with (172) to determine $z_{0}(y)$. We also note that, in general,
(181) will not satisfy the boundary conditions in (174d), and passive boundary
layers will be needed, when $y=O(\widetilde{\epsilon}^{\frac{1}{2}})^{+}$ and
$y=1-O(\widetilde{\epsilon}^{\frac{1}{2}})^{+}$. For brevity, we do not
consider these boundary layers here. We observe from (181) that,

$0<\psi(y)\leq 1~{}~{}\forall~{}~{}y\in[0,1]$ (182)

as specified in the normalisation (174c). In addition the stationary points of
$\alpha(y)$ and $\psi(y)$ are coincident. Now, equation (174a) can be re-
written as

$\left((\psi(y)z_{0}(y))^{\prime}\right)^{2}+\psi^{2}(y)-1=0,~{}~{}y\in(0,1).$
(183)

Therefore,

$(\psi(y)z_{0}(y))^{\prime}=\begin{cases}\left(1-\psi^{2}(y)\right)^{\frac{1}{2}},~{}~{}0\leq
y<y_{M},\\\ -\left(1-\psi^{2}(y)\right)^{\frac{1}{2}},~{}~{}y_{M}\leq y\leq
1,\end{cases}$ (184)

so that an integration gives,

$z_{0}(y)=c\psi^{-1}(y)+\psi^{-1}(y)\begin{cases}\int_{0}^{y}{\left(1-\psi^{2}(s)\right)^{\frac{1}{2}}}ds,~{}~{}0\leq
y<y_{M},\\\
\int_{0}^{y_{M}}{\left(1-\psi^{2}(y)\right)^{\frac{1}{2}}}ds-\int_{y_{M}}^{y}{\left(1-\psi^{2}(y)\right)^{\frac{1}{2}}}ds,~{}~{}y_{M}\leq
y\leq 1,\end{cases}$ (185)

with the constant $c$ determined by the condition (179). We note that, as
required, $z_{0}\in C^{1}([0,1])$, with passive boundary layers required when
$y,(y-1)=O(\widetilde{\epsilon}^{\frac{1}{2}})$ over which the boundary
conditions (174d) will be satisfied. In summary we have, in this case of
balanced advection, with the channel width large compared to the streamwise
diffusion length scale (based on the reaction time scale), a PTW solution for
each $A>0$, with propagation speed

$\hat{v}(A,B,u_{c})=v^{\ast}(u_{c})+A\alpha_{M}+O(B^{\frac{1}{2}})$ (186)

as $B\to 0$ with $A=O(1)$. In addition, the moving boundary has location,

$z=B^{-\frac{1}{2}}z_{0}(y)+O(1),~{}~{}y\in[0,1],$ (187)

as $B\to 0$ with $A=O(1)$, $z_{0}:[0,1]\to\mathbb{R}$ given by (185) and
$\psi:[0,1]\to\mathbb{R}$ given by (174b). We now compute the interface
deformation for the Poiseuille flow (8b) by numerically integrating (185) with
(179) for three different values of $v^{*}(u_{c})$ corresponding to small,
intermediate and large cut-off $u_{c}$. As the cut-off increases, the maximum
of the interface and its deformation increases.

##  6 Conclusions

In this paper we have considered a reaction-diffusion process evolving inside
an infinite channel in the presence of a shear flow. The reaction function is
of standard KPP-type, but experiences a cut-off in the reaction rate below the
normalised cut-off concentration $u_{c}\in(0,1)$. We have formulated this
initial-value boundary problem (IBVP) in terms of an equivalent moving
boundary evolution problem (QIVP) and examined both of these problems in a
number of natural asymptotic limits relating to the two non-dimensional
(positive) parameters $A$ and $B$ describing the strength of the flow and
front thickness, whilst keeping $u_{c}$ fixed. We have established in all
cases considered, that a unique (up to spatial translation) PTW solution
$U_{T}(x-vt,y)=u(x,y,t)$ exists, and in a number of cases, that this forms a
large-$t$ attractor for (IBVP). We used the method of matched asymptotic
expansions to determine the detailed structure of the PTW solutions and their
speed of propagation and, in particular, their dependence on the parameters.
It is of interest to examine the nature of the propagation speed
$\hat{v}(A,B,u_{c})$ in the three complementary regimes $A\to\infty$, $A=O(1)$
and $A\to 0$ over the full range of $B$. Using the results captured by Figure
3, $\hat{v}(A,B,u_{c})$ is fully determined, up to the stated orders in $A$,
as a function of $B$ as $A\to 0$. Partial results obtained in the other two
regimes are completed by extracting and extrapolating limiting forms of the
available asymptotic expressions. We therefore expect that for $A=O(1)$ and
$A\to\infty$, the speed of propagation decreases monotonically with $B$. We
anticipate that the approach developed in this paper will be readily adaptable
to the case of two corresponding problems, when the KPP-type cut-off reaction
function is replaced by a broader class of cut-off reaction functions or when
the shear flow is unsteady, varying periodically with time. Finally, it would
be interesting to determine whether similar qualitative effects arise for a
certain stochastically perturbed KPP equation obtained from an alternative
model whose purpose is also to account for microscopic discrete particles [6,
7]. Mueller, Mytnik and Quastel [20] have shown that in the absence of a
shear, the difference between the speed obtained from this model and
$v^{*}(u_{c})$ obtained from the deterministic cut-off model considered here
is small when the cut-off $u_{c}$ and noise are both small. Whether this
difference continues to be small in the presence of a shear flow is presently
unknown.

Acknowledgments. The authors acknowledge A. D. O. Tisbury for his early
contribution on this project. We also thank the referees for their
constructive comments, which have led to an overall improvement in the
presentation of the paper.

##  Appendix A Asymptotic solution to (QIVP) as $t\to 0^{+}$

As $t\to 0^{+},$ the nature of the discontinuity in the initial condition of
(QIVP) and the requirement for this to be locally spatially smoothed in the
small time limit, demands a diffusion balance at leading order in (QIVP). As a
consequence of this, it follows that the solution to (QIVP) develops in four
principal asymptotic regions, which are,

  * •

Region $\textrm{I}_{L}$ : $\xi^{\prime}=O(t^{\frac{1}{2}})^{-}$,
$y^{\prime}=O(1)$ with $u=O(1)$ as $t\to 0^{+}$

  * •

Region $\textrm{I}_{R}$ : $\xi^{\prime}=O(t^{\frac{1}{2}})^{+}$,
$y^{\prime}=O(1)$ with $u=O(1)$ as $t\to 0^{+}$

  * •

Region $\textrm{II}_{L}$ : $\xi^{\prime}=O(1)^{-}$, $y^{\prime}=O(1)$ with
$u=1-o(1)$ as $t\to 0^{+}$

  * •

Region $\textrm{II}_{R}$ : $\xi^{\prime}=O(1)^{+}$, $y^{\prime}=O(1)$ with
$u=o(1)$ as $t\to 0^{+}$

with, in addition,

$s(t)=o(1)~{}~{},~{}~{}\zeta(y^{\prime},t)=o(1)~{}~{}\text{as}~{}~{}t\to
0^{+}.$ (188)

A similar structure characterises the case in the absence of advection and
this is developed in [25], where a qualitative sketch of the above regions is
also included. Throughout $O(\cdot)^{\pm}$ denotes $O(\cdot)$ being positive
or negative, respectively. A balance of terms in the PDE (QIVPa) indicates
that, with regions $\textrm{I}_{L,R}$ having $\xi=O(t^{\frac{1}{2}})$ as $t\to
0^{+}$, then,

$s(t)=O(t^{\frac{1}{2}}),~{}~{}\zeta(y^{\prime},t)=O(t^{\frac{1}{2}})$

as $t\to 0^{+}$. Thus we expand in the form,

$s(t)=s_{0}t^{\frac{1}{2}}+s_{1}t+O(t^{\frac{3}{2}})\quad\text{and}\quad\zeta(y^{\prime},t)=\zeta_{0}(y^{\prime})t^{\frac{1}{2}}+\zeta_{1}(y^{\prime})t+O(t^{\frac{3}{2}})$
(189)

as $t\to 0^{+}$, with $y^{\prime}\in[0,L]$. The details in regions
$\textrm{I}_{L,R}$ are straightforward, obtained by introducing the coordinate
$\eta=\xi^{\prime}t^{-1/2}$ and in both regions expanding

$u(\eta,y^{\prime},t)=u_{0}(\eta,y^{\prime})+t^{1/2}u_{1}(\eta,y^{\prime})+O(t),\quad\text{as}\quad
t\to 0^{+}.$ (190)

Substituting expansions (189), and (190) into the equation for $u$ we obtain
that $u_{0}=u_{0}(\eta)$ and satisfies

$u_{0\eta\eta}+\frac{1}{2}(\eta+s_{0})u_{0\eta}=0,~{}~{}\eta\in\mathbb{R}\setminus\\{0\\}$
(191a) subject to $u_{0}(0)=u_{c},\quad u_{0\eta}(0^{-})=u_{0\eta}(0^{+})$
(191b) together with matching conditions with regions IIL and IIR which
requires, $u_{0}(\eta)\to\begin{cases}1\quad\text{as}\quad\eta\to-\infty,\\\
0\quad\text{as}\quad\eta\to\infty.\end{cases}$ (191c)

It is straightforward to determine

$u_{0}(\eta)=\frac{1}{2}\text{erfc}(\frac{1}{2}(\eta+s_{0})),\quad\text{where}\quad
s_{0}=2\text{erfc}^{-1}(2u_{c})\quad\text{with}\quad\zeta_{0}(y^{\prime})=0,$
(192)

for $(\eta,y^{\prime})\in\mathbb{R}\times[0,L]$. At the next order, we obtain
that $u_{1}=u_{1}(\eta,y^{\prime})$ satisfies

$u_{1\eta\eta}+\frac{1}{2}(\eta+s_{0})u_{1\eta}=A\bar{\alpha}(y^{\prime}),~{}~{}(\eta,y^{\prime})\in(\mathbb{R}\setminus\\{0\\})\times(0,L)$
(193a) subject to
$u_{1}(0,y^{\prime})=-\zeta_{1}(y^{\prime})u_{0\eta}(0),\quad
u_{1\eta}(0^{-},y^{\prime})=u_{1\eta}(0^{+},y^{\prime}),$ (193b) with
$y^{\prime}\in[0,L]$, together with matching conditions with regions IIL and
IIR which requires, $u_{1}(\eta,y)\to
0\quad\text{as}\quad|\eta|\to\infty~{}~{}\text{uniformly for}~{}~{}y\in[0,L].$
(193c)

Solving (193) yields

$u_{1}(\eta,y^{\prime})=\frac{\zeta_{1}(y^{\prime})}{2\sqrt{\pi}}e^{-\frac{1}{4}(\eta+s_{0})^{2}},\quad\text{where}\quad
s_{1}=0,\quad\text{with}\quad\zeta_{1}(y^{\prime})=A\bar{\alpha}(y^{\prime}),$
(194)

for $(\eta,y^{\prime})\in\mathbb{R}\times[0,L]$. Thus, we have via (192) and
(194) that

$u(\eta,y^{\prime},t)=\frac{1}{2}\text{erfc}(\frac{1}{2}(\eta+s_{0}))+\frac{A\bar{\alpha}(y^{\prime})}{2\sqrt{\pi}}e^{-\frac{1}{4}(\eta+s_{0})^{2}}t^{\frac{1}{2}}+O(t)$
(195a) as $t\to 0^{+}$, uniformly for
$(\eta,y^{\prime})=(O(1)^{\pm})\times[0,L]$, with
$s(t)=2\text{erfc}^{-1}(2u_{c})t^{\frac{1}{2}}+O(t^{\frac{3}{2}})$ (195b) and
$\zeta(y^{\prime},t)=A\bar{\alpha}(y^{\prime})t+O(t^{\frac{3}{2}})$ (195c)
uniformly for $y^{\prime}\in[0,L]$ as $t\to 0^{+}$.

An interesting interpretation of these asymptotic forms is that $s(t)$ gives
the diffusive displacement, whilst $\zeta(y^{\prime},t)$ gives the advective
displacement, respectively, of the spatial moving boundary in the early times.
However, reaction plays no role, up to $O(t^{\frac{3}{2}})$.

We now move on to regions $\mathrm{II}_{L,R}$. It follows from (195a) that $u$
and $u-1$ are exponentially small in $t$ as $t\to 0^{+}$ in regions
$\mathrm{II}_{R}$ and $\mathrm{II}_{L}$, respectively. The details are
lengthy, but standard (see, for example, [25] for more details for the case
when $\bar{\alpha}(y^{\prime})=0$ for $y^{\prime}\in[0,L]$), and are omitted
for brevity. After matching with regions $\mathrm{I}_{L}$ and $\mathrm{I}_{R}$
respectively, we have

$u(\xi^{\prime},y^{\prime},t)=H(-\xi^{\prime})+\text{sgn}(\xi^{\prime})\exp{(-\phi(\xi^{\prime},y^{\prime},t)t^{-1})}$
(196)

with $H(\cdot)$ being the usual Heaviside function, and

$\phi(\xi^{\prime},y^{\prime},t)=\frac{1}{4}\xi^{\prime
2}+\frac{1}{2}s_{0}\xi^{\prime}t^{\frac{1}{2}}-\left(\frac{1}{2}\log{t}-\frac{1}{4}s_{0}^{2}-\frac{1}{2}\log{\pi}+\frac{1}{2}A\bar{\alpha}(y^{\prime})\xi^{\prime}-\log(|\xi^{\prime}|)\right)t+O(t^{\frac{3}{2}})$
(197)

as $t\to 0^{+}$, with $(\xi^{\prime},y^{\prime})\in(O(1)^{\pm})\times[0,L]$.

At this stage the asymptotic structure as $t\to 0^{+}$ is not quite complete.
In obtaining the expansions in regions $\mathrm{I}_{L,R}$ and
$\mathrm{II}_{L,R}$, we have been forced to neglect the Neumann boundary
conditions at $y^{\prime}=0$ and $y^{\prime}=L$, and an examination of these
expansions reveals that,

$u_{y^{\prime}}\sim O(t^{\frac{1}{2}})$ (198)

when $y^{\prime}=0,L$ in regions $\mathrm{I}_{L,R}$ as $t\to 0^{+}$, whilst,

$u_{y^{\prime}}\sim O(e^{-\phi_{L,R}t^{-1}})$ (199)

when $y^{\prime}=0,L$ in regions $\mathrm{II}_{L,R}$ as $t\to 0^{+}$. Thus,
_weak_ and _passive_ boundary layers are required when
$y^{\prime}=O(t^{\frac{1}{2}})$ and $y^{\prime}=L-O(t^{\frac{1}{2}})$ as $t\to
0^{+}$, adjacent to both regions $\mathrm{I}_{L,R}$ and $\mathrm{II}_{L,R}$.
These boundary layers are readily dealt with, and since they are passive in
nature, for brevity we do not present details here. This completes the
asymptotic structure of the solution to (QIVP) as $t\to 0^{+}$.

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