ref1ref1footnotetext: This work was supported by Project MTM2013-42907 DGICT
Spain

# Title

Mihaela Negreanu<EMAIL_ADDRESS>Departamento de Matemática Aplicada,
Universidad Complutense de Madrid, 28040 Madrid, Spain  J. Ignacio Tello
<EMAIL_ADDRESS>Departamento de Matemática Aplicada a las tecnologías de la
Información y las Telecomunicaciones, E.T.S.I. Sistemas Informáticos.
Universidad Politécnica de Madrid, 28031 Madrid, Spain
Center for Computational Simulation, Universidad Politécnica de Madrid, 28660
Boadilla del Monte, Madrid, Spain

# On a Parabolic-Elliptic system with gradient dependent chemotactic
coefficient

Mihaela Negreanu<EMAIL_ADDRESS>Departamento de Matemática Aplicada,
Universidad Complutense de Madrid, 28040 Madrid, Spain  J. Ignacio Tello
<EMAIL_ADDRESS>Departamento de Matemática Aplicada a las tecnologías de la
Información y las Telecomunicaciones, E.T.S.I. Sistemas Informáticos.
Universidad Politécnica de Madrid, 28031 Madrid, Spain
Center for Computational Simulation, Universidad Politécnica de Madrid, 28660
Boadilla del Monte, Madrid, Spain

###### Abstract

We consider a second order PDEs system of Parabolic-Elliptic type with
chemotactic terms. The system describes the evolution of a biological species
“$u$” moving towards a higher concentration of a chemical stimuli “$v$” in a
bounded and open domain of ${{{\rm l}\kern-1.49994pt{\rm R}}}^{N}$. In the
system considered, the chemotaxis sensitivity depends on the gradient of $v$,
i.e., the chemotaxis term has the following expression

$-div\left(\chi u|\nabla v|^{p-2}\nabla v\right),$

where $\chi$ is a positive constant and $p$ satisfies

$p\in(1,\infty),\quad\mbox{ if }N=1\quad\mbox{ and }\quad
p\in\left(1,\frac{N}{N-1}\right),\quad\mbox{ if }N\geq 2.$

We obtain uniform bounds in time in $L^{\infty}(\Omega)$ of the solutions. For
the one-dimensional case we prove the existence of infinitely many non-
constant steady-states for $p\in(1,2)$ for any $\chi$ positive and a given
positive mass.

Keywords: Chemotaxis, Global Existence of solutions, infinitely many solutions

##  1 Introduction

Chemotaxis is the ability of some living organisms to orient their movement
along a chemical concentration gradient. The process has been extensively
studied from a biological point of view after the development of the
microscope during the XIX century. In the last decades, several mathematical
models have been presented to describe the phenomenon, after the pionnering
works of Patlak [29] and Keller and Segel [23] (see also the review articles
Horstmann [19], [20] and Bellomo et al [3] and references therein for more
extensive literature in the subject). The original model in [23] describes the
evolution of a biological species, denoted by “$u$” in terms of a parabolic
equation, with linear diffusion and a second order nonlinear term in the form

$-div(\chi u\nabla v),$

where $v$ denotes the concentration of the chemical stimuli.
In the last years, linear diffusion of the biological especies ”$u$” has been
replaced in different ways:

  * 1.

by nonlinear diffusion at the form $-div(\phi(u)\nabla u)$, see for instance
Wrzosek [39], Cieslak and Mortales-Rodrigo [16], Cieslak and Winkler [17],
Winkler [38].

  * 2.

by fractional diffusion, see J Burczak, R Granero-Belinchón [13] and [14]
among others.

  * 3.

by nonlinear diffusion depending on $|\nabla u|^{p}$, (p-laplatian), see
Bendahmane [6].

The system has been also studied for several biological species, see for
instance Tang and Tao [33], Tello and Winkler [36], Stinner, Tello and Winkler
[32], Negreanu and Tello [27] and [28], Wang and Wu [37] among others. In the
last years, several mathematical models have considered the chemotactic
sensitivity coefficient “$\chi$” dependent of $\nabla v$ instead of constant.
For instance, in Bellomo and Winkler [4] and [5] (see also Bellomo et al [2]),
a chemotaxis system is analyzed for a chemotactic term of the form

$-div\left(\frac{\chi u}{[1+|\nabla v|^{2}]^{\frac{1}{2}}}\nabla v\right).$

In Bianchi, Painter and Sherratt [7] the authors consider the term

$-div\left(\frac{\chi u}{(1+\omega u)}\frac{\nabla v}{(1+\eta|\nabla
v|)}\right),$

for some positive constants $\chi$, $\omega$ and $\eta$. In [7], the authors
study a system of four PDEs of parabolic type in an one-dimensional spatial
domain coupled with an ODE modeling Lymphangiogenesis in wound healing.
A general chemotactic term is presented as

$-div\left[u\tilde{\chi}(u,v,|\nabla v|)\nabla v\right],$

where $\tilde{\chi}$ is a continuos function for $|\nabla v|>0$.
In the present work we consider a simplified case, where $\tilde{\chi}$ is
given by

$\tilde{\chi}(u,v,|\nabla v|)=\chi|\nabla v|^{p-2},$

for some positive constant $\chi$ and $p>1$.
We study the problem in a bounded spatial open domain $\Omega\subset{{{\rm
l}\kern-1.49994pt{\rm R}}}^{N}$, with regular boundary $\partial\Omega$ and
denote by $\vv{n}$ the outward pointing normal vector on the boundary
$\partial\Omega$. The equation for $v$ is restricted to the elliptic case, for
simplicity, we assume that $v$ satisfies the Poisson equation and the system
studied is the following

$u_{t}-\Delta u=-div(\chi u|\nabla v|^{p-2}\nabla v),\quad\quad
x\in\Omega,\quad t>0,$ (1.1) $-\Delta
v=u-M,\quad\quad\qquad\qquad\qquad\quad\;\;\;\;x\in\Omega,\quad t>0,$ (1.2)

with Neumann boundary conditions

$\displaystyle\frac{\partial u}{\partial\vv{n}}=\displaystyle\frac{\partial
v}{\partial\vv{n}}=0,\qquad\qquad\qquad\qquad\qquad\quad
x\in\partial\Omega,\quad t>0$ (1.3)

and a non-negative initial data

$u(0,x)=u_{0}(x),\qquad\qquad\qquad\quad x\in\Omega,$ (1.4)

satisfying

$\frac{1}{|\Omega|}\displaystyle\int_{\Omega}u_{0}(x)dx=M.$

Notice that if $(u,v)$ is a solution, $(u,v+k)$ is also a solution for any
constant $k$. To obtain uniqueness of the problem, we impose a given mass for
the species $v$, i.e., we assume

$\int_{\Omega}v=0.$ (1.5)

The problem for $p=2$ has been already analyzed by different authors, starting
with the work of Jäger and Luckhaus [22] and Biler [8], and [9], see also
Nagai [25], Senba [31], Naito and Suzuki [26], Blanchet, Dolbeault and
Perthame [11], Blanchet, Carrillo and Masmoundi [10] among others.

In this article we analyze the case where $p$ satisfies

$p\in(1,\infty),\quad\mbox{ if }N=1\quad\mbox{ and }\quad
p\in\left(1,\frac{N}{N-1}\right),\quad\mbox{ if }N\geq 2,$ (1.6)

assuming initial data

$u_{0}\in C^{2,\alpha}(\overline{\Omega}),\quad\mbox{ for some
$\alpha\in(0,1)$ }$ (1.7)

and

$\frac{\partial u_{0}}{\partial\vv{n}}=0,\quad x\in\partial\Omega.$ (1.8)

In Section 2 we study the global existence of the solutions $(u,v)$ of system
(1.1)-(1.8). The main result is enclosed in the following theorem.

###### Theorem 1.1

Under assumptions (1.5)-(1.8), for any $T<\infty$, there exists an unique
classical solution to (1.1)-(1.4),

$u,v\in C^{2+\alpha,1+\frac{\alpha}{2}}_{x,t}(\overline{\Omega}_{T}).$

Moreover, there exists a constant $C(u_{0},\chi,p,\Omega)$, independent of
$T$, such that

$\|u\|_{L^{\infty}(\Omega)}\leq C.$

In Section 3 we consider the steady states of the problem (1.1) - (1.5) and we
prove the existence of infinitely many solutions in one dimensional bounded
domain for $p\in(1,2)$.

##  2 Global existence of Solutions

To prove the global existence of solutions we apply Schauder Fixed Point
Theorem. We first introduce the local existence results in Lemma 2.1 and
obtain a priori estimates presented in the subsequent lemmas.

###### Lemma 2.1

Under assumptions (1.6)-(1.8), there exists a unique solution $(u,v)$ to
(1.1)-(1.4) in $(0,T_{max})$ satisfying

$u,v\in C^{2+\alpha,1+\frac{\alpha}{2}}_{x,t}(\Omega_{T}),\qquad\mbox{ for any
$T<T_{max}$}$

where $T_{max}$ is a positive number satisfying

$\limsup_{t\rightarrow
T_{max}}\left(\|u(t)\|_{L^{\infty}(\Omega)}+\|v(t)\|_{L^{\infty}(\Omega)}+t\right)=\infty.$
(2.1)

Moreover, the solution $u$ satisfies

$u(t,x)\geq 0,\qquad x\in\Omega,\quad t<T_{max}.$ (2.2)

Proof. For any $T<T_{max}$ we have that $u$ satisfies

$u_{t}-\Delta u+b(t,x)\cdot\nabla u=f(x,t),\quad(t,x)\in\Omega_{T},$

where

$b(x,t)=\chi|\nabla v|^{p-2}\nabla v,\qquad f(x,t)=div\left(\chi|\nabla
v|^{p-2}\nabla v\right).$

Since $u\in L^{\infty}(\Omega_{T})$ we have that $v\in
L^{s}(0,T:W^{2,q}(\Omega))$ for any $s,q\geq 1$ and therefore

$b(x,t)\in L^{s}(0,T:W^{1,q}(\Omega)),\qquad f(x,t)\in
L^{s}(0,T:L^{q}(\Omega)).$

Then, $u\in C^{2+\alpha,1+\frac{\alpha}{2}}_{x,t}(\Omega_{T})$ see Remark 48.3
(ii) in Quittner-Souplet [30].

The non-negativity of $u$ is a consequence of the maximum principle.

Notice that, after integration in (1.1), we have that the total mass is
preserved in time, i.e.,

$\int_{\Omega}u=\int_{\Omega}u_{0}=|\Omega|M.$ (2.3)

###### Remark 2.2

Let $\Omega\subset{{{\rm l}\kern-1.49994pt{\rm R}}}^{N}$, be a bounded and
regular domain and $f\in L^{1}(\Omega)$, such that $\int_{\Omega}f=0$, then,
the problem

$\left\\{\begin{array}[]{l}\displaystyle\displaystyle-\Delta v=f,\qquad\mbox{
in }\Omega\\\ \\\ \displaystyle\frac{\partial
v}{\partial\vv{n}}=0,\quad\quad\;\;\mbox{ in
}\partial\Omega,\end{array}\right.$

has an unique solution

$v\in W^{1,s}(\Omega),\quad\mbox{ for
}\quad\left\\{\begin{array}[]{l}s\in[1,N/(N-1)),\mbox{ if }N>1\\\\[5.69054pt]
s=\infty\mbox{ if }N>1,\end{array}\right.$ (2.4)

satisfying

$\int_{\Omega}v=0.$

The proof of (2.4) is given in Chabroswki [15] Theorem 2.8, where the general
problem

$\left\\{\begin{array}[]{l}\displaystyle-\Delta v=\lambda v+f,\qquad\mbox{ in
}\Omega\\\ \\\ \displaystyle\frac{\partial
v}{\partial\vv{n}}=0,\quad\qquad\qquad\;\mbox{ in
}\partial\Omega,\end{array}\right.$

is studied for $\lambda\in{{{\rm l}\kern-1.49994pt{\rm R}}}$.

###### Lemma 2.3

Let $p>1$ such that

$\left\\{\begin{array}[]{l}p<\infty,\quad\quad\quad\quad\quad\mbox{ if }\quad
N=1,\\\\[8.53581pt] 2(p-1)<\frac{N}{N-1},\quad\mbox{ if }\quad N\geq
2,\end{array}\right.$ (2.5)

then, for any $q>1$ and any $s>0$ satisfying

$\left\\{\begin{array}[]{c}s\in\left(2(p-1),\infty\right),\quad\quad\;\mbox{
if }\quad N=1,\\\\[5.69054pt]
s\in\left(2(p-1),\frac{N}{N-1}\right),\quad\mbox{ if }\quad N\geq
2,\end{array}\right.$ (2.6)

the following inequality holds:

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{3(q-1)}{q}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq
c_{1}q(q-1)\chi^{2}\left[\int_{\Omega}u^{\frac{sq}{s-2(p-1)}}\right]^{\frac{s-2(p-1)}{s}},$
(2.7)

for some positive constant $c_{1}$.

Proof: We multiply equation (1.1) by $u^{q-1}$ (for $q>1$) and integrate by
parts to obtain

$\displaystyle\frac{d}{dt}\frac{1}{q}\int_{\Omega}u^{q}+(q-1)\int_{\Omega}\left|\nabla
u\right|^{2}u^{q-2}=(q-1)\chi\int_{\Omega}\left|\nabla v\right|^{p-2}\nabla
v\nabla u\cdot u^{q-1}.$ (2.8)

Thanks to Young’s inequality, it results

$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\frac{1}{q}\int_{\Omega}u^{q}+\frac{4(q-1)}{q^{2}}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\displaystyle\leq\frac{2(q-1)\chi}{q}\int_{\Omega}\left|\nabla
v\right|^{p-1}\left|\nabla u^{\frac{q}{2}}\right|u^{\frac{q}{2}}\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle&\leq\displaystyle\frac{(q-1)}{q^{2}}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}+(q-1)\chi^{2}\int_{\Omega}\left|\nabla
v\right|^{2p-2}u^{q}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle&\displaystyle\leq\frac{(q-1)}{q^{2}}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}+(q-1)\chi^{2}\left[\int_{\Omega}\left|\nabla
v\right|^{s}\right]^{\frac{2(p-1)}{s}}\left[\int_{\Omega}u^{\frac{sq}{s-2(p-1)}}\right]^{\frac{s-2(p-1)}{s}},\end{array}$

then

$\frac{d}{dt}\frac{1}{q}\int_{\Omega}u^{q}+\frac{3(q-1)}{q^{2}}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\displaystyle\leq(q-1)\chi^{2}\left[\int_{\Omega}\left|\nabla
v\right|^{s}\right]^{\frac{2(p-1)}{s}}\left[\int_{\Omega}u^{\frac{sq}{s-2(p-1)}}\right]^{\frac{s-2(p-1)}{s}}.$
(2.9)

We notice that, since $u\in L^{1}(\Omega)$, we have $v\in W^{1,s}(\Omega)$ for
any $s$ satisfying (2.6) and, thanks to Lemma 2.4, we have

$\|\nabla v\|_{L^{s}(\Omega)}^{2(p-1)}\leq c_{1}<\infty.$ (2.10)

Then, equation (2.9) becomes

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{3(q-1)}{q}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq\chi^{2}(q-1)qc_{1}\left[\int_{\Omega}u^{\frac{sq}{s-2(p-1)}}\right]^{\frac{s-2(p-1)}{s}},$

the proof is done.

###### Lemma 2.4

Under assumption

$\left\\{\begin{array}[]{l}p\in(1,\infty),\qquad\quad\;\;\quad\mbox{ if
}N=1,\\\\[8.53581pt] p\in\left(1,\frac{N}{N-1}\right),\qquad\quad\mbox{ if
}N\geq 2,\end{array}\right.$ (2.11)

and any $s>0$ satisfying

$\left\\{\begin{array}[]{c}s\in\left(2(p-1),\infty\right),\quad\qquad\mbox{ if
}\quad N=1,\\\\[5.69054pt] s\in\left(2(p-1),\frac{N}{N-1}\right),\quad\;\mbox{
if }\quad N\geq 2,\end{array}\right.$ (2.12)

we have that

$\int_{\Omega}u^{N+1}\leq c_{5},\qquad\mbox{ for any }t>0,$ (2.13)

and $c_{5}$ independent of $t$.

Proof. We first notice that for $N\geq 2$ we have that $p<2$ and then $p\geq
2(p-1)$ and, in view of (2.11),

$2(p-1)\leq p<\frac{N}{N-1}$

which gives a non empty set of admissible values of $s$ for $N\geq 2$.
To prove the Lemma we follow an Moser-Alikakos iteration method, the result
could be also obtained using similar arguments as in Tao and Winkler [34], for
readers convenience we detail the proof. We recall the expression (2.7)

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{3(q-1)}{q}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq\chi^{2}(q-1)qc_{1}\left[\int_{\Omega}u^{\frac{sq}{s-2(p-1)}}\right]^{\frac{s-2(p-1)}{s}}.$

By Gagliardo-Nirenberg inequality (see Henry [18]), for
$1\leq\gamma\leq\beta\leq\infty$, $r\geq 1$ and

$-\frac{N}{\beta}=a(1-\frac{N}{r})-\frac{N}{\gamma}(1-a),$

if

$a\in(0,1),$

we have that

$\|w\|_{L^{\beta}(\Omega)}\leq
C_{GN}\|w\|_{L^{\gamma}(\Omega)}^{1-a}\|w\|_{W^{1,r}(\Omega)}^{a}.$ (2.14)

Notice that

$a=\frac{\frac{1}{\gamma}-\frac{1}{\beta}}{\frac{1}{\gamma}+\frac{1}{N}-\frac{1}{r}}.$

We take

$w=u^{\frac{q}{2}},\quad\gamma=1\qquad\beta=\frac{2s}{s-2(p-1)},\quad
r=2,\quad\mbox{ and }\ s=\frac{N}{N-1+\epsilon}$

for $\epsilon$ small enough such that

$s\in\left(2(p-1),\frac{N}{N-1}\right),\quad N-\frac{p}{p-1}+\epsilon<0.$
(2.15)

Then, we obtain

$\|u^{\frac{q}{2}}\|_{L^{\frac{2s}{s-2(p-1)}}(\Omega)}\leq
C_{GN}\|u^{\frac{q}{2}}\|_{H^{1}(\Omega)}^{a}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{1-a},$

for

$\begin{array}[]{ll}\displaystyle
a&=\displaystyle\frac{1-\frac{s-2(p-1)}{2s}}{\frac{1}{2}+\frac{1}{N}}=\frac{2N}{N+2}(1-\frac{s-2(p-1)}{2s})=\frac{N}{N+2}\frac{(s+2(p-1))}{s}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle&\displaystyle=\frac{N}{N+2}\frac{(\frac{N}{N-1+\epsilon}+2(p-1))}{\frac{N}{N-1+\epsilon}},\end{array}$

equivalent to

$\begin{array}[]{ll}\displaystyle
a&=\displaystyle\frac{N-1+\epsilon}{N+2}\left(\frac{N}{N-1+\epsilon}+2(p-1)\right)=\frac{N+2(p-1)(N-1+\epsilon)}{N+2}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle&\displaystyle<\frac{N+2+2(p-1)N-2p+2(p-1)\epsilon}{N+2}=1+2(p-1)\frac{N-\frac{p}{p-1}+\epsilon}{N+2}.\end{array}$

Notice that

$N<\frac{p}{p-1}\quad\quad\quad\mbox{ $\Longleftrightarrow$ }\quad\quad\quad
p<\frac{N}{N-1},$

and in view of (2.15) we have $a<1$.

We now apply inequality (2.14) to the last term in (2.7) to obtain

$\left[\int_{\Omega}u^{\frac{sq}{s-2(p-1)}}\right]^{\frac{s-2(p-1)}{s}}\leq
C_{GN}\left[\|u^{\frac{q}{2}}\|_{H^{1}(\Omega)}^{a}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{1-a}\right]^{2}$

and

$C_{GN}\left[\|u^{\frac{q}{2}}\|_{H^{1}(\Omega)}^{a}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{1-a}\right]^{\frac{s-2(p-1)}{s}}\leq\delta\|u^{\frac{q}{2}}\|_{H^{1}(\Omega)}^{2}+\frac{c_{2}}{\delta^{\frac{a}{(1-a)}}}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{2},$

for some positive $\delta$ small enough. Using the above estimates, (2.7)
becomes

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{3(q-1)}{q}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq\chi^{2}(q-1)qc_{1}\left[\delta\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}+\delta\int_{\Omega}u^{q}+\frac{c_{2}}{\delta^{\frac{a}{(1-a)}}}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{2}\right],$

i.e.,

$\frac{d}{dt}\int_{\Omega}u^{q}+\left[\frac{3(q-1)}{q}-\delta\chi^{2}(q-1)qc\right]\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq\chi^{2}(q-1)qc\left[\delta\int_{\Omega}u^{q}+\frac{c_{2}}{\delta^{\frac{a}{(1-a)}}}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{2}\right]$

and for $\delta<\frac{2}{\chi^{2}cq^{2}}$ it results

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{(q-1)}{q}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq\chi^{2}(q-1)qc\left[\delta\int_{\Omega}u^{q}+\frac{c_{2}}{\delta^{\frac{a}{(1-a)}}}\|u^{\frac{q}{2}}\|_{L^{1}(\Omega)}^{2}\right].$
(2.16)

Thanks to Poincaré-Wirtinger inequality we get

$\int_{\Omega}\left|\nabla u^{\frac{q}{2}}\right|^{2}\geq
C_{PW}\int_{\Omega}\left(u^{\frac{q}{2}}-\frac{1}{|\Omega|}\int_{\Omega}u^{\frac{q}{2}}\right)^{2}=C_{PW}\left(\int_{\Omega}u^{q}-\frac{1}{|\Omega|}\left|\int_{\Omega}u^{\frac{q}{2}}\right|^{2}\right),$

which implies

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{q-1}{q}\left[\frac{C_{PW}}{2}-\delta
q^{2}c\chi^{2}\right]\int_{\Omega}u^{q}\leq
c_{3}(\delta,|\Omega|,q)\left|\int_{\Omega}u^{\frac{q}{2}}\right|^{2}.$

For $\delta<\min\\{\frac{C_{PW}}{4c\chi^{2}q^{2}},\frac{2}{\chi^{2}cq^{2}}\\}$
the last inequality is reduced to

$\frac{d}{dt}\int_{\Omega}u^{q}+\frac{q-1}{q}\frac{C_{PW}}{4}\int_{\Omega}u^{q}\leq
c_{3}(\delta,|\Omega|,q)\left|\int_{\Omega}u^{\frac{q}{2}}\right|^{2}$ (2.17)

and by the Maximum Principle, we get

$\sup_{t>0}\|u\|_{L^{q}(\Omega)}^{q}\leq\frac{4qc_{3}(\delta,|\Omega|,q)}{(q-1)C_{PW}}\sup_{t>0}\|u\|^{q}_{L^{\frac{q}{2}}(\Omega)}.$
(2.18)

Following Moser-Alikakos iteration (see [1]), we define

$x_{i}:=\sup_{t>0}\int_{\Omega}u^{2^{i}},\qquad i\in{{{\rm
l}\kern-1.49994pt{\rm N}}}.$

Thanks to (2.3) it results $x_{0}=|\Omega|M<\infty$ and (2.18) implies that

$x_{i}\leq c_{4}(\delta,i)x_{i-1}^{2}.$

Notice that $x_{i}$ is finite for $i<\infty$, in particular, there exists
$i_{0}$ large enough, such that $2^{i_{0}}>N+1$ and therefore

$\int_{\Omega}u^{N+1}\leq c_{5}<\infty\quad\mbox{ for any }t>0,$

and the proof ends.

###### Lemma 2.5

There exists a positive constant $c_{6}$, independent of $t$, such that the
following bound holds

$\|\nabla v\|_{L^{\infty}(\Omega)}<c_{6}.$

Proof: In view of (2.13) we claim that $v\in W^{2,N+1}(\Omega)\subset
W^{1,\infty}(\Omega)$, thanks to Chabroswki [15] Theorem 2.8, we have the
result.

###### Lemma 2.6

There exists a positive constant $c_{\infty}$, independent of $t$, such that

$\|u\|_{L^{\infty}(\Omega)}\leq c_{\infty}.$

Proof: From (2.8) (for $q\geq 2$) and Lemma 2.5 we have

$\frac{d}{dt}\frac{1}{q}\int_{\Omega}u^{q}+(q-1)\int_{\Omega}\left|\nabla
u\right|^{2}u^{q-2}=(q-1)c_{7}\int_{\Omega}|\nabla u|u^{q-1}.$

Since

$c_{7}\int_{\Omega}|\nabla u|u^{q-1}\leq\frac{1}{2}\int_{\Omega}\left|\nabla
u\right|^{2}u^{q-2}+\frac{c_{7}^{2}}{2}\int_{\Omega}u^{q},$

we get

$\frac{d}{dt}\frac{1}{q}\int_{\Omega}u^{q}+\frac{(q-1)}{2}\int_{\Omega}\left|\nabla
u\right|^{2}u^{q-2}=(q-1)\frac{c_{7}^{2}}{2}\int_{\Omega}u^{q}.$ (2.19)

In the Gagliardo-Nirenberg’s inequality

$\|w\|_{L^{2}(\Omega)}\leq C_{GN1}\|\nabla
w\|^{a}_{L^{2}(\Omega)}\|w\|^{1-a}_{L^{1}(\Omega)}+C_{GN2}\|w\|_{L^{1}(\Omega)},\quad\mbox{
for }\quad\frac{1}{2}=a\left(\frac{1}{2}-\frac{1}{N}\right)+1-a,$

taking

$w=u^{\frac{q}{2}},\qquad\mbox{ for }\quad a=\frac{N}{N+2}<1,$

we obtain

$\left[\int_{\Omega}u^{q}\right]^{\frac{1}{2}}\leq
C_{GN1}\left[\int_{\Omega}|\nabla
u^{\frac{q}{2}}|^{2}\right]^{\frac{a}{2}}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{(1-a)}+C_{GN2}\int_{\Omega}u^{\frac{q}{2}},$

which is equivalent to

$\int_{\Omega}u^{q}\leq 2C_{GN1}\left[\int_{\Omega}|\nabla
u^{\frac{q}{2}}|^{2}\right]^{a}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2(1-a)}+2C_{GN2}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2}.$

Replacing the last expression in (2.19) we have

$\frac{d}{dt}\frac{1}{q}\int_{\Omega}u^{q}+\frac{2(q-1)}{q^{2}}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq(q-1)c_{8}\left(\left[\int_{\Omega}|\nabla
u^{\frac{q}{2}}|^{2}\right]^{a}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2(1-a)}+\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2}\right)$
(2.20)

and applying Young’s inequality to the first term of the right side,

$\left[\int_{\Omega}|\nabla
u^{\frac{q}{2}}|^{2}\right]^{a}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2(1-a)}\leq\frac{1}{q^{2}c_{8}}\int_{\Omega}|\nabla
u^{\frac{q}{2}}|^{2}+\left[q^{2}c_{8}a\right]^{\frac{a}{(1-a)}}(1-a)\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2},$

inequality (2.20) is reduced to

$\frac{1}{q}\frac{d}{dt}\int_{\Omega}u^{q}+\frac{(q-1)}{q^{2}}\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq(q-1)q^{\frac{2a}{(1-a)}}c_{9}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2},$

i.e.,

$\frac{q}{(q-1)}\frac{d}{dt}\int_{\Omega}u^{q}+\int_{\Omega}\left|\nabla
u^{\frac{q}{2}}\right|^{2}\leq
q^{\frac{2}{(1-a)}}c_{9}\left[\int_{\Omega}u^{\frac{q}{2}}\right]^{2}$

Thanks to Poincaré-Wirtinger’s inequality we have

$\int_{\Omega}\left|\nabla u^{\frac{q}{2}}\right|^{2}\geq
C_{PW}\int_{\Omega}\left(u^{\frac{q}{2}}-\frac{1}{|\Omega|}\int_{\Omega}u^{\frac{q}{2}}\right)^{2}=C_{PW}\left(\int_{\Omega}u^{q}-\frac{1}{|\Omega|}\left|\int_{\Omega}u^{\frac{q}{2}}\right|^{2}\right),$

which implies

$\frac{q}{q-1}\frac{d}{dt}\int_{\Omega}u^{q}+C_{PW}\int_{\Omega}u^{q}\leq
q^{2+\frac{2a}{(1-a)}}c_{10}\left|\int_{\Omega}u^{\frac{q}{2}}\right|^{2}.$

By the Maximum Principle we get

$\sup_{t>0}\|u\|_{L^{q}(\Omega)}^{q}\leq\frac{q^{\frac{2}{(1-a)}}c_{10}}{C_{PW}}\sup_{t>0}\|u\|^{q}_{L^{\frac{q}{2}}(\Omega)}.$
(2.21)

As in Lemma 2.4 we apply the following Moser-Alikakos iteration (see [1]),
defining

$x_{i}:=\sup_{t>0}\int_{\Omega}u^{2^{i}},\qquad i\in{{{\rm
l}\kern-1.49994pt{\rm N}}}.$

Thanks to (2.3) we have that $x_{0}=|\Omega|M<\infty$ and (2.18) implies that

$x_{i}\leq\displaystyle 2^{\frac{2i}{(1-a)}}c_{11}x_{i-1}^{2},$

then

$x_{i}^{2^{-i}}\leq\displaystyle
2^{\frac{i2^{1-i}}{(1-a)}}c_{11}^{2^{-i}}x_{i-1}^{2^{1-i}},$

and

$\displaystyle\|u\|_{L^{2^{i}}(\Omega)}\leq\displaystyle\prod_{j=0\dots
i}2^{\displaystyle\frac{j2^{1-j}}{(1-a)}}\displaystyle
c_{11}^{2^{-j}}=\displaystyle 2^{\displaystyle\sum_{j=0\dots
i}\displaystyle\frac{j2^{1-j}}{(1-a)}}c_{11}^{\displaystyle\sum_{j=0\dots
i}2^{-j}}.$

Since

$\begin{array}[]{ll}\displaystyle\sum_{j=0\dots
i}\frac{j2^{1-j}}{(1-a)}&=\displaystyle\frac{2}{(1-a)}\sum_{j=0\dots
i}j2^{-j}=\frac{2}{(1-a)}(2-2^{-i}-2^{-i-1})\leq\frac{4}{(1-a)}\end{array}$

and

$\sum_{j=0\dots i}2^{-j}=2-2^{-i}<2,$

we obtain

$\|u\|_{L^{2^{i}}(\Omega)}\leq 2^{\frac{2}{(1-a)}}c_{11}^{2}:=c_{\infty}.$

Constant $c_{11}$ is independent of $i$, taking limits when
$i\rightarrow\infty$, the proof ends.
End of the proof of Theorem 1.1.
In view of Lemma 2.5 and assumption 1.5 we obtain uniform boundedness of $v$
in $L^{\infty}(\Omega_{T_{max}})$. Thanks to Lemmatta 2.1 and 2.6 the proof
ends.

###### Remark 2.7

We notice that, if equation (1.2) is replaced by

$-\Delta v+v=u,\quad\quad x\in\Omega,\quad t>0,$ (2.22)

and Remark 2.2 is replaced by Lemma 23 in Brezis-Strauss [12], we also obtain
the boundedness in $L^{\infty}(\Omega)$ of $u$ and $v$ and the global
existence of solutions.

##  3 Stationary states in 1 D

The steady states of the problem (1.1)-(1.5) for $\Omega=(0,1)$, are given by

$\left\\{\begin{array}[]{ll}-u_{xx}=-(\chi
u|v_{x}|^{p-2}v_{x})_{x},&x\in(0,1),\\\ -v_{xx}=u-M,&x\in(0,1),\\\
u_{x}(0)=u_{x}(1)=v_{x}(0)=v_{x}(1)=0.&\end{array}\right.$ (3.1)

The aim of this section is to prove the following theorem:

###### Theorem 3.1

Let $p\in(1,2)$, then, for any $M>0$ and $\chi>0$, there exist infinitely many
solutions to (3.1).

In order to prove the theorem, we proceed into several steps.
We consider the following variables

$w:=ln(u)-ln(M),\qquad\beta:=v_{x}$ (3.2)

then, (3.1) becomes

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
w_{x}=\chi|\beta|^{p-2}\beta,&x\in(0,1),\\\
\displaystyle\beta_{x}=M(1-e^{w}),&x\in(0,1),\\\
\beta(0)=\beta(1)=0.&\end{array}\right.$ (3.3)

We have that (3.3) is a Hamiltonian system, i.e., there exists $H(w,\beta)$
such that

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\chi|\beta|^{p-2}\beta=\frac{\partial
H(w,\beta)}{\partial\beta},&x\in(0,1)\\\ \displaystyle
M(1-e^{w})=-\frac{\partial H(w,\beta)}{\partial
w},&x\in(0,1).\end{array}\right.$ (3.4)

After integration, we obtain

$H(w,\beta)=M(e^{w}-w-1)+\chi\frac{|\beta|^{p}}{p}.$ (3.5)

The solutions of the system are found along the contours of $H$, so that, to
sketch the phase diagram of this system, it is enough to study and draw the
level sets of the Hamiltonian function $H$. We notice that there exists an
energy in the system (3.3) which is preserved along the solutions, i.e.,

$\frac{M}{\chi}(e^{w}-w-1)+\frac{1}{p}|\beta|^{p}=k,$ (3.6)

where $k$ is a non negative constant. Notice that for $k=0$ we have the
trivial solution $w=\beta=0$.
In the following lemma we prove the conservation of the energy.

###### Lemma 3.2

Let $(w,\beta)$ be a solution to (3.3) and the hamiltonian function $H$ as in
(3.5), i.e.,

$H(w,\beta)=M(e^{w}-w-1)+\frac{\chi}{p}|\beta|^{p}$

then,

$\frac{dH}{dx}=0.$

Proof: Recalling the Hamilton function $H$ is conserved in any solution of the
system, i.e.,

$\frac{dH(w(x),\beta(x))}{dx}=\frac{\partial H}{\partial w}\frac{\partial
w}{\partial x}+\frac{\partial H}{\partial\beta}\frac{\partial\beta}{\partial
x}=\frac{\partial H}{\partial w}\frac{\partial
H}{\partial\beta}-\frac{\partial H}{\partial\beta}\frac{\partial H}{\partial
w}=0,$

and we have the proof.
Notice that the level sets of $H$ are bounded curves corresponding to a
periodic solution $(w,\beta)$ of period $T(k)$, for $k$ defined in (3.6).

End of proof of Theorem 3.1.
We write $|\beta|$ in terms of $|w_{x}|$, then

$|\beta|^{p}=\chi^{\frac{p}{1-p}}|w_{x}|^{\frac{p}{p-1}}$

and

$\frac{M}{\chi}(e^{w}-w-1)+\frac{\chi^{\frac{p}{1-p}}}{p}|w_{x}|^{\frac{p}{p-1}}=k$
(3.7)

and $|w_{x}|$ satisfies

$|w_{x}|=[kp\chi^{\frac{p}{p-1}}+Mp\chi^{\frac{1}{p-1}}(1+w-e^{w})]^{\frac{p-1}{p}}.$
(3.8)

We denote by $r_{0}$ and $-r_{1}$ the values of $w$ for $\beta=0$ for a given
$k$, then $r_{1}$ and $r_{0}$ satisfy

$k=\frac{M}{\chi}(e^{r_{0}}-r_{0}-1)=\frac{M}{\chi}(e^{-r_{1}}+r_{1}-1).$

After integration in (3.8) we get

$\int_{-r_{1}}^{r_{0}}\left[kp\chi^{\frac{p}{p-1}}+Mp\chi^{\frac{1}{p-1}}(1+w-e^{w})\right]^{\frac{1-p}{p}}dw=\frac{T}{2},$

i.e.

$\int_{-r_{1}}^{r_{0}}\left[\frac{k\chi}{M}+(1+w-e^{w})\right]^{\frac{1-p}{p}}dw=[Mp\chi^{\frac{1}{p-1}}]^{\frac{p-1}{p}}\frac{T}{2},$
(3.9)

and

$\int_{-r_{1}}^{r_{0}}\left[\frac{k\chi}{M}+(1+w-e^{w})\right]^{\frac{p-1}{p}}dw=\int_{-r_{1}}^{0}\left[e^{-r_{1}}+r_{1}-e^{w}+w\right]^{-\frac{p-1}{p}}dw$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{0}^{r_{0}}\left[e^{r_{0}}-r_{0}-e^{w}+w\right]^{-\frac{p-1}{p}}dw.$

To estimate the righthand side integrals, we apply Hölder inequality, so, for
any $\epsilon>0$ verifying

$\epsilon<\frac{2-p}{p-1}$ (3.10)

we have:

$\int_{0}^{r_{0}}\left[e^{r_{0}}-r_{0}-e^{w}+w\right]^{-\frac{p-1}{p}}dw\leq\int_{0}^{r_{0}}(e^{w}-1)\left[e^{r_{0}}-r_{0}-e^{w}+w\right]^{-\frac{(p-1)(2+\epsilon)}{p}}dw+\int_{0}^{r_{0}}(e^{w}-1)^{-\frac{1}{1+\epsilon}}.$

After integration, for $p<2$, we get

$\int_{0}^{r_{0}}(e^{w}-1)\left[e^{r_{0}}-r_{0}-e^{w}+w\right]^{-\frac{(p-1)(2+\epsilon)}{p}}dw=\frac{p}{(2+\epsilon)-p(1+\epsilon)}\left[e^{r_{0}}-r_{0}-1\right]^{\frac{(2+\epsilon)-p(1+\epsilon)}{p}}$

and

$\int_{0}^{r_{0}}(e^{w}-1)^{-\frac{1}{1+\epsilon}}\leq\int_{0}^{r_{0}}w^{-\frac{1}{1+\epsilon}}=\frac{1+\epsilon}{\epsilon}r_{0}^{\frac{\epsilon}{1+\epsilon}}.$

In the same way we obtain

$\int_{-r_{1}}^{0}\left[e^{-r_{1}}+r_{1}-e^{w}+w\right]^{-\frac{p-1}{p}}dw\leq\int_{-r_{1}}^{0}(1-e^{w})\left[e^{-r_{1}}+r_{1}-e^{w}+w\right]^{-\frac{(p-1)(2+\epsilon)}{p}}dw$
$\qquad\qquad+\int_{-r_{1}}^{0}(1-e^{w})^{-\frac{1}{1+\epsilon}}dw.$

Computing the above integrals it follows

$\int_{-r_{1}}^{0}(1-e^{w})\left[e^{-r_{1}}+r_{1}-e^{w}+w\right]^{-\frac{(p-1)(2+\epsilon)}{p}}dw=\frac{p}{(2+\epsilon)-p(1+\epsilon)}\left[e^{-r_{1}}+r_{1}-1\right]^{1-\frac{(p-1)(2+\epsilon)}{p}}$

and

$\int_{-r_{1}}^{0}(1-e^{w})^{-\frac{1}{1+\epsilon}}dw=\int_{0}^{r_{1}}(1-e^{-w})^{-\frac{1}{1+\epsilon}}dw.$

In wiew of

$1-e^{-w}\geq\frac{1-e^{-r_{1}}}{r_{1}}x,\quad x\in(0,r_{1})$

it results

$\int_{0}^{r_{1}}(1-e^{-w})^{-\frac{1}{1+\epsilon}}dw\leq(1-e^{-r_{1}})^{-\frac{1}{1+\epsilon}}r_{1}^{\frac{1}{1+\epsilon}}\int_{0}^{r_{1}}\frac{1}{x^{\frac{1}{1+\epsilon}}}dw=\frac{1+\epsilon}{\epsilon}(1-e^{-r_{1}})^{-\frac{1}{1+\epsilon}}r_{1}.$

Therefore

$[Mp\chi^{\frac{1}{p-1}}]^{\frac{p-1}{p}}\frac{T}{2}\leq\left[\frac{p}{(2+\epsilon)-p(1+\epsilon)}\left[e^{r_{0}}-r_{0}-1\right]^{\frac{(2+\epsilon)-p(1+\epsilon)}{p}}+\frac{1+\epsilon}{\epsilon}r_{0}^{\frac{\epsilon}{1+\epsilon}}\right.$
$\left.+\frac{p}{(2+\epsilon)-p(1+\epsilon)}\left[e^{-r_{1}}+r_{1}-1\right]^{\frac{(2+\epsilon)-p(1+\epsilon)}{p}}+\frac{1+\epsilon}{\epsilon}(1-e^{-r_{1}})^{-\frac{1}{1+\epsilon}}r_{1}\right]$

which implies

$\lim_{k\rightarrow 0}T=\lim_{r_{1},r_{0}\rightarrow 0}T=0,\qquad\mbox{ for
}p\in(1,2).$

To determine $\displaystyle\lim_{k\rightarrow\infty}T$ we apply the definition
of $T$

$T=2[Mp\chi^{\frac{1}{p-1}}]^{\frac{1-p}{p}}\int_{-r_{1}}^{r_{0}}[\frac{\chi}{M}k-(1+w-e^{w})]^{\frac{1-p}{p}}dw.$

The following bound holds

$\displaystyle T\geq
C\int_{0}^{\frac{r_{1}}{2}}[r_{1}+e^{-r_{1}}-w-e^{-w})]^{\frac{1-p}{p}}dw\geq
C\frac{r_{1}}{2}\min_{w\in(0,r_{1}/2)}\left[r_{1}+e^{-r_{1}}-w-e^{-w}\right]^{\frac{1-p}{p}}$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad=C\frac{r_{1}}{2}\left[\frac{r_{1}}{2}+e^{-r_{1}}-e^{-r_{1}/2}\right]^{\frac{1-p}{p}},$

and for $r_{1}>2$ we have that

$\frac{r_{1}}{2}+e^{-r_{1}}-e^{-r_{1}/2}\leq\frac{r_{1}}{2}+1\leq r_{1}.$

Applying it in the previous inequality that verifies $T$, we get

$\displaystyle T\geq cr_{1}^{\frac{1}{p}}$

and then

$\lim_{k\rightarrow\infty}T=\lim_{r_{1}\rightarrow\infty}T=\infty,\qquad\mbox{
for }p\in(1,2).$

By the continuity of $T$ with respect to $k$, there exist $k_{n}>0$ such that
$T_{n}$, the period of the corresponding solution $(w_{n},\beta_{n})$ to the
energy constant $k_{n}$, satisfies

$T_{n}=\frac{1}{n}$

and therefore

$w_{n}^{\prime}(0)=w_{n}^{\prime}(1)=0.$

We recover the solution of the original problem $(u_{n},v_{n})$ defined by

$u_{n}(x)=Me^{w_{n}(x)},\qquad
v_{n}(x)=\int_{0}^{x}\beta_{n}(s)ds-\int_{0}^{1}\beta_{n}(s)ds$

and the proof ends.

###### Remark 3.3

We notice that, the steady states for the 1-dimensional problem (1.2)-(1.4),
defined by (3.1), are equivalent to the solutions of the problem

$\left\\{\begin{array}[]{ll}\displaystyle-
u_{xx}=-\chi(uw_{x})_{x},&x\in\Omega,\\\\[5.69054pt]
\displaystyle-(|w_{x}|^{\frac{2-p}{p-1}}w_{x})_{x}=u-M,&x\in\Omega,\\\\[5.69054pt]
\displaystyle w_{x}(0)=w_{x}(1)=0.&\end{array}\right.$

Since $u=Me^{\chi w}$, we have

$\left\\{\begin{array}[]{ll}\displaystyle-(|w_{x}|^{\frac{2-p}{p-1}}w_{x})_{x}=M\left(e^{\chi
w}-1\right),&x\in\Omega,\\\\[5.69054pt] \displaystyle
w_{x}(0)=w_{x}(1)=0,&\end{array}\right.$

for $w$ satisfying

$\int_{\Omega}e^{\chi w}=1.$

## Acknowledgments

This work is supported by the Project MTM2017-83391-P from MICINN (Spain).

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