X^{l}F\in\mathcal{M}_{+}\text{ for some $l>0$}\\}.\end{split}$ (4.27)

If in addition $\mathcal{R}\cap\mathbb{R}^{d}_{<l}=\varnothing$ for some
$l>0$, then $\mathcal{X}_{\mathcal{R}}=\mathscr{D}^{\prime}_{\mathcal{R}}$.

In particular in the periodic case, $\mathcal{R}$ can not contain any point on
the boundary $\xi_{1}=0$ as mentioned above, so the last statement holds.
(4.27) applies to $\mathcal{X}$ itself if $d=1$.

###### Proof.

In view of Lemma 2.4, it suffices to derive the condition (X-ii) from the
other conditions. For any $l\in(0,\infty)$, the boundedness of
$\mathcal{R}\cap\mathbb{R}^{d}_{<l}$ with the definition of
$\mathscr{D}^{\prime}$ implies that $F$ is of finite order on
$\mathbb{R}^{d}_{<l}$. Hence there exists $m(l)\in\mathbb{N}$ such that
$|F(\varphi)|\leq m(l)\|\varphi\|_{m(l)}=m(l)\|\varphi\|_{m(l)}^{+}$ for all
$\varphi\in\mathscr{D}_{(0,l)}$, which is enough for the condition (X-ii), by
the proof of Lemma 2.4. The last sentence of lemma is obvious. ∎

Examples of $\mathcal{R}$ satisfying the local boundedness are conical regions
in the form

$\begin{split}\mathcal{R}:=\\{\xi\in\mathbb{R}^{d}:|\xi^{\perp}|\leq
R(\xi_{1})\\},\end{split}$ (4.28)

for some $R\in C^{+}_{0}$ that is super-additive ($R(a)+R(b)\leq R(a+b)$ for
all $a,b\geq 0$), which implies $\mathcal{R}+\mathcal{R}\subset\mathcal{R}$.
The left side $|\xi^{\perp}|$ may be replaced with any semi-norm on
$\mathbb{R}^{d}$ that bounds $\xi^{\perp}$. We may choose $R$ to grow
arbitrarily fast by

######  Lemma 4.4.

For any $d\in\mathbb{N}$, any non-empty $A\subset\mathbb{R}^{d}_{\geq 0}$ and
any locally bounded $R_{0}:A\to[0,\infty)$ satisfying

$\begin{split}\limsup_{\xi\in A,\
|\xi|\to+0}|\xi|^{-1}R_{0}(\xi)<\infty,\end{split}$ (4.29)

there exists a super-additive $R\in C^{\scaleto{\nearrow}{4pt}}_{0}$
satisfying $R_{0}(\xi)\leq R(|\xi|)$ for all $\xi\in A$.

###### Proof.

Let $R_{1}(l):=\sup\\{|\eta|^{-1}R_{0}(\eta)\mid\eta\in A,\ 0<|\eta|\leq
l\\}$. Then $R_{1}:(0,\infty)\to[0,\infty)$ by the assumptions, and it is non-
decreasing. Then $R:[0,\infty)\to[0,\infty)$ defined by
$R(l):=\int_{l}^{2l}R_{1}(t)dt$ belongs to $C^{\scaleto{\nearrow}{4pt}}_{0}$,
satisfying

$\begin{split}R(a+b)\geq\int_{a+b}^{2a+b}R_{1}(t-b)dt+\int_{2a+b}^{2a+2b}R_{1}(t-2a)dt=R(a)+R(b)\end{split}$
(4.30)

and $R(|\xi|)\geq|\xi|R_{1}(|\xi|)\geq R_{0}(\xi)$ for all $\xi\in A$. ∎

######  Remark 4.1.

In addition to the translation invariance, the space
$\mathcal{F}^{-1}\mathcal{X}$ is invariant for the exponential Fourier
multipliers in the form
$e^{z\partial_{1}}:=\mathcal{F}^{-1}e^{iz\xi_{1}}\mathcal{F}$ for
$z\in\mathbb{C}$, which is the holomorphic extension for the $x_{1}$ variable.
Indeed, the multiplication with $e^{iz\xi_{1}}$ is bounded
$\mathcal{X}^{\mu,\nu_{0}}\to\mathcal{X}^{\mu,\nu_{1}}$ if
$e^{|\mathop{\mathrm{Im}}z|l}\nu_{0}(l)\leq\nu_{1}(l)$ for all $l>0$, so it is
continuous on $\mathcal{X}$. Moreover, the convolution is invariant: for any
$F,G\in\mathcal{X}$ and $z\in\mathbb{C}$, we have

$\begin{split}e^{iz\xi_{1}}(F*G)=(e^{iz\xi_{1}}F)*(e^{iz\xi_{1}}G),\end{split}$
(4.31)

which is immediate for $F,G\in\mathscr{D}(\mathbb{R}^{d})$ and then extended
by the density of $\mathscr{D}_{>0}\hookrightarrow\mathcal{X}$, Proposition
2.6. Hence the set of solutions has the same invariance: If $u\in
C(I;(\mathcal{F}^{-1}\mathcal{X})^{n_{1}})$ is a solution of (1.1) on an
interval $I$, then $e^{z\partial_{1}}u\in
C(I;(\mathcal{F}^{-1}\mathcal{X})^{n_{1}})$ is also a solution for any
$z\in\mathbb{C}$.

##  5\. Comparison with other solutions

Although our definition of solutions is quite natural in view of the Fourier
transform of the equation, it is non-trivial to compare those solutions with
others defined and constructed in different ways. The bottleneck to use the
uniqueness in $\mathcal{X}$ is the Fourier support condition and decay at
$\xi_{1}=0$.

Nevertheless, the following is a natural procedure to compare our solutions in
$\mathcal{X}$ with some others, when it is easy to obtain the latter. Suppose

  1. (1)

Global wellposedness in $\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$ holds
(Corollary 4.2) for some $\mathcal{R}\subset\mathbb{R}^{d}$ such that the
closure of the interior of $\mathcal{R}$ contains $0$.

  2. (2)

Local wellposedness holds in another space $X$ such that
$\mathscr{D}_{>0}\hookrightarrow\mathcal{F}X\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})$.

  3. (3)

$I_{n}$ in Proposition 2.6 is an approximating sequence also on $X$.

Then we may choose the approximating sequence $I_{n}$ of Proposition 2.6 such
that $I_{n}:\mathcal{X}_{\mathcal{R}}\to\mathscr{D}_{\mathcal{R}}$. For any
initial data $u_{0}\in(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}}$, we
have an approximating sequence
$u_{0,n}:=\mathcal{F}^{-1}I_{n}\hat{u}_{0}\in(\mathcal{F}^{-1}\mathscr{D}_{\mathcal{R}})^{n_{1}}\hookrightarrow
X^{n_{1}}$. Let $u_{n}$ be the solution for the initial data $u_{0,n}$ for
each $n\in\mathbb{N}$. The global wellposedness in
$\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$ implies that $u_{n}\to u$ in
$C([0,\infty);(\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}})^{n_{1}})$. The local
wellposedness in $X$ implies that $u_{n}\in C([0,T_{n});X^{n_{1}})$ for the
maximal existence time $T_{n}\in(0,\infty]$. If $u_{0}\in X^{n_{1}}$, then we
have $u_{0,n}\to u_{0}$ by the approximation property in $X$, so the local
wellposedness in $X$ implies $u_{n}\to u$ in $C([0,T);X^{n_{1}})$, where
$T\in(0,\infty]$ is the maximal time of $u$. Here the limit $u$ is the same,
as long as the convergence holds, because both of $\mathcal{X}$ and
$\mathcal{F}X$ are continuously embedded into
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$.

Therefore, the solutions are the same between
$\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$ and $X$, for all the initial data
in their intersection as long as the solution exists in $X$. In this sense, we
may regard the solutions in $\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$ as
extension from $X\cap\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$. However,
unless the global wellposedness holds also in $X$, the maximal existence time
$T_{n}$ may well go to $0$ as $n\to\infty$ when $u_{0}\not\in X^{n_{1}}$.

###  5.1. Solutions beyond blow-up

Blow-up may well happen in finite time in $X$, while the solution is extended
globally in $\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$. A typical example is
the ODE

$\begin{split}\dot{u}=u^{2},\end{split}$ (5.1)

whose Cauchy problem is explicitly solved with the Taylor expansion:

$\begin{split}u(t)=\tfrac{u_{0}}{1-tu_{0}}=\sum_{n\in\mathbb{N}_{0}}t^{n}u_{0}^{n+1},\end{split}$
(5.2)

whose radius of convergence is $|tu_{0}|<1$ with the simple pole at
$tu_{0}=1$. The iteration sequence $\\{v_{n}\\}_{n\geq 0}$ defined by

$\begin{split}\dot{v}_{n}=v_{n-1}^{2},\quad v_{0}(0)=u_{0},\end{split}$ (5.3)

is in the form (which may be shown by induction)

$\begin{split}v_{n}(t)=\sum_{0\leq k\leq
2^{n}-1}c_{n,k}t^{k}u_{0}^{k+1},\quad\begin{cases}c_{n,k}=1&(0\leq k\leq n)\\\
0<c_{n,k}<1&(n<k\leq 2^{n}-1).\end{cases}\end{split}$ (5.4)

Using $\|f*g\|_{\mathcal{M}}\leq\|f\|_{\mathcal{M}}\|g\|_{\mathcal{M}}$ for
the Fourier transform of the equation, it is easy to see that this equation is
locally wellposed in $\mathcal{F}^{-1}\mathcal{M}$, and the iteration
converges to $u$ for $|t|\|u_{0}\|_{\mathcal{F}^{-1}\mathcal{M}}<1$, where
$\|\varphi\|_{\mathcal{F}^{-1}\mathcal{M}}:=\|\mathcal{F}\varphi\|_{\mathcal{M}}$.
Moreover, if $\hat{u}_{0}$ is a non-negative measure, then we have
$\|\hat{u}_{0}*\hat{u}_{0}\|_{\mathcal{M}}=\|\hat{u}_{0}\|_{\mathcal{M}}^{2}$
and similarly for iterations, so

$\begin{split}\|u(t)\|_{\mathcal{F}^{-1}\mathcal{M}}=\sum_{n\in\mathbb{N}_{0}}t^{n}\|u_{0}\|_{\mathcal{F}^{-1}\mathcal{M}}^{n+1}=\tfrac{\|u_{0}\|_{\mathcal{F}^{-1}\mathcal{M}}}{1-t\|u_{0}\|_{\mathcal{F}^{-1}\mathcal{M}}}\end{split}$
(5.5)

blows up exactly as $t\|u_{0}\|_{\mathcal{F}^{-1}\mathcal{M}}\to 1-0$. Since
the non-negative Fourier transform implies
$\|u(t)\|_{L^{\infty}}=u(t,0)=\|u(t)\|_{\mathcal{F}^{-1}\mathcal{M}}$, it also
blows up in $L^{\infty}_{x}$ and at $x=0$, as well as in smaller Banach
algebras, such as
$H^{s}(\mathbb{R}^{d})\hookrightarrow\mathcal{F}^{-1}L^{1}\hookrightarrow\mathcal{F}^{-1}\mathcal{M}$
for $s>d/2$. On the other hand, $u(t)=\sum_{n=0}^{\infty}t^{n}u_{0}^{n+1}$ is
convergent and smooth for all $t\in\mathbb{R}$ in $\mathcal{X}$. All the above
remains valid when $\mathcal{M}$ is replaced with the closed subspace
$L^{1}(\mathbb{R}^{d})$.

A more interesting case is the Burgers equation

$\begin{split}u_{t}=u_{xx}+2uu_{x}\end{split}$ (5.6)

for $u(t,x):\mathbb{R}^{1+1}\to\mathbb{C}$, which is well-known to be
explicitly solvable by the Cole-Hopf transform, thereby the blow-up has been
intensively studied, cf. [30] for the complex-valued blow-up solutions on
$\mathbb{R}$. A simple example within our function space is for the
monochromatic initial data

$\begin{split}u_{0}(x)=-iae^{ix},\end{split}$ (5.7)

with a parameter $a\in\mathbb{C}$, for which [19] proved blow-up in
$L^{2}(\mathbb{R})$. Here we investigate the unique global solution $u\in
C(\mathbb{R};\mathcal{F}^{-1}\mathcal{X})$ (given by Corollary 4.2), using the
explicit formula in the Fourier space as in [19]:

$\begin{split}&u(t)=\sum_{n\in\mathbb{N}}u^{n}(t)e^{inx},\quad
u^{1}(t)=-iae^{-t},\\\ &u^{n}(t)=in\int_{0}^{t}e^{-n^{2}(t-s)}\sum_{1\leq
k\leq n-1}u^{k}(s)u^{n-k}(s)ds\quad(n\geq 2).\end{split}$ (5.8)

Changing the variable from $u^{n}$ to $U_{n}$ defined by

$\begin{split}u^{n}(t)=-i(ae^{-t})^{n}U_{n}(t),\end{split}$ (5.9)

the equation may be rewritten as

$\begin{split}U_{1}(t)=1,\quad
U_{n}(t)=\int_{0}^{t}e^{-n(n-1)(t-s)}n\sum_{1\leq k\leq
n-1}U_{k}(s)U_{n-k}(s)ds.\end{split}$ (5.10)

Then we easily observe by induction on $n\in\mathbb{N}$ that

$\begin{split}U_{2}(t)^{n-1}=(1-e^{-2t})^{n-1}\leq U_{n}(t)\leq 1.\end{split}$
(5.11)

Indeed, it is obviously satisfied for $n=1,2$, while for $n\geq 3$, the upper
bound is immediate by injecting the induction hypothesis for $k=1,\ldots,n-1$
into the equation. For the lower bound, the same injection yields

$\begin{split}U_{n}(t)\geq
e^{-n(n-1)t}\int_{0}^{t}n(n-1)e^{n(n-1)s}U_{2}(s)^{n-2}ds,\end{split}$ (5.12)

and the conclusion follows by replacing the integrand with

$\begin{split}\partial_{s}[e^{n(n-1)s}U_{2}(s)^{n-1}]&=n(n-1)e^{n(n-1)s}U_{2}(s)^{n-2}(1-\tfrac{n-2}{n}e^{-2s})\\\
&\leq n(n-1)e^{n(n-1)s}U_{2}(s)^{n-2}.\end{split}$ (5.13)

For the original Fourier coefficients, we thus obtain

$\begin{split}|a|e^{-t}\cdot|ae^{-t}(1-e^{-2t})|^{n-1}\leq|u^{n}(t)|=|ae^{-t}|^{n}U_{n}(t)\leq|ae^{-t}|^{n}.\end{split}$
(5.14)

Note that $e^{-t}(1-e^{-2t})$ attains its maximum $\tfrac{2}{3\sqrt{3}}$ at
$t=\tfrac{1}{2}\log 3$.

On the other hand, the equation is locally wellposed in
$\mathcal{F}^{-1}\mathcal{M}$, where the above solution $u$ satisfies

$\begin{split}\sum_{n\in\mathbb{N}}|u^{n}(t)|=\|u(t)\|_{\mathcal{F}^{-1}\mathcal{M}}=iu(t,x_{a})=\|u(t)\|_{L^{\infty}_{x}},\end{split}$
(5.15)

for all $x_{a}\in\mathbb{R}$ satisfying $ae^{ix}=|a|$. The classical blow-up
occurs when the above diverges. Hence the threshold on $|a|$ for blow-up is
given by

$\begin{split}&\mathfrak{B}:=\\{(t,a)\in[0,\infty)^{2}\mid\sum_{n\in\mathbb{N}}(ae^{-t})^{n}U_{n}(t)=\infty\\},\\\
&a_{*}(t):=\inf\\{a\mid(t,a)\in\mathfrak{B}\\},\quad a_{**}:=\inf_{t\geq
0}a_{*}(t).\end{split}$ (5.16)

(5.14) implies

$\begin{split}e^{t}\leq a_{*}(t)\leq\tfrac{e^{t}}{1-e^{-2t}},\quad 1\leq
a_{**}\leq\tfrac{3}{2}\sqrt{3}=\inf_{t\geq
0}\tfrac{e^{t}}{1-e^{-2t}},\end{split}$ (5.17)

while the local wellposedness in $\mathcal{F}^{-1}\mathcal{M}$ implies
$\lim_{t\to+0}a_{*}(t)=\infty$. The classical blow-up time is given by

$\begin{split}T_{*}(a):=T_{*}(|a|)=\inf\\{t>0\mid
a_{*}(t)\leq|a|\\}>0\quad(\inf\varnothing=\infty).\end{split}$ (5.18)

Indeed, for $0\leq t<T_{*}(a)$, $|a|<a_{*}(t)$ implies that $u^{n}(t)$ is
exponentially decaying as $n\to\infty$, so $u(t)$ is analytic. If
$u(T_{*}(a))\in\mathcal{F}^{-1}\mathcal{M}$, then the local wellposedness in
$\mathcal{F}^{-1}\mathcal{M}$ implies that $u\in
C([0,T];\mathcal{F}^{-1}\mathcal{M})$ for some $T>T_{*}(a)$, and also for
slightly bigger $|a|$ by the continuous dependence on $u(0)$, contradicting
the definition of $T_{*}(a)$. Hence, if $T_{*}(a)<\infty$, then
$(T_{*}(a),|a|)\in\mathfrak{B}$ and
$T_{*}(a)=\min\\{t\mid(t,|a|)\in\mathfrak{B}\\}$, so $a_{*}(T_{*}(a))\leq|a|$,
while $T_{*}(a_{*}(t))\leq t$ is obvious by definition. Then (5.14) implies

$\begin{split}e^{T_{*}(a)}\leq|a|\leq\tfrac{e^{T_{*}(a)}}{1-e^{-2T_{*}(a)}}.\end{split}$
(5.19)

In particular $T_{*}(a)\leq\log|a|$. It is obvious by definition that
$T_{*}(a)=\infty$ for $|a|<a_{**}$ and $T_{*}(a)<\infty$ for $|a|>a_{**}$. Let
$T_{**}:=T_{*}(a_{**})$. If $T_{**}=\infty$ then by the local wellposedness as
above, the solution with slightly bigger $|a|$ exists for $t<\log a_{**}+1$,
contradicting $T_{*}(a)\leq\log|a|$ for $|a|>a_{**}$. Hence
$(T_{**},a_{**})\in\mathfrak{B}$, $a_{*}(T_{**})=a_{**}=\min a_{*}$, and $u$
blows up iff $|a|\geq a_{**}$, and at $t=T_{*}(a)\leq\log|a|$. Since $T_{*}$
is non-increasing by definition, the blow-up time is bounded for all
$a\in\mathbb{C}$ by

$\begin{split}\max\\{T_{*}(a)\mid a\in\mathbb{C},\
T_{*}(a)<\infty\\}=T_{*}(a_{**})\leq\log|a_{**}|.\end{split}$ (5.20)

If $a_{*}(t)<|a|$ (which happens iff $|a|>a_{**}$), $u^{n}(t)$ is
exponentially divergent as $n\to\infty$ (along a subsequence) so
$u(t)\not\in\mathscr{S}^{\prime}(\mathbb{R})$. There can be some time gap
after $T_{*}(a)$ if $a_{*}(t)\geq|a|$ on some time interval starting from
$T_{*}(a)$, but such intervals are mutually disjoint, so those exceptional
$|a|$ are at most countable. Thus we obtain

######  Proposition 5.1.

There exist unique $a_{**}\in(1,\tfrac{3}{2}\sqrt{3}]$,
$T_{**}\in(0,\log\tfrac{3}{2}\sqrt{3})$ and
$a_{*}:(0,\infty)\to[a_{**},\infty)$, satisfying
$\lim_{t\to+0}a_{*}(t)=\infty$, (5.17) and $\min a_{*}=a_{**}=a_{*}(T_{**})$,
such that the unique solution $u\in C(\mathbb{R};\mathcal{F}^{-1}\mathcal{X})$
of (5.6)–(5.7) blows up in the classical sense if and only if $|a|\geq a_{**}$
and at $T_{*}(a):=\min a_{*}^{-1}((0,|a|])\leq\min(T_{**},\log|a|)$. For those
$t>0$ where $|a|<a_{*}(t)$, $u(t)$ is analytic. For $|a|>a_{*}(t)$, $u^{n}(t)$
is exponentially divergent as $n\to\infty$ and so
$u(t)\not\in\mathscr{S}^{\prime}(\mathbb{R})$. Excepting a countable set of
$|a|$, $u(t)$ gets out of $\mathscr{S}^{\prime}(\mathbb{R})$ immediately after
$T_{*}(a)$, while $u(t)$ is analytic for $t<T_{*}(a)$ and for
$t>T^{*}(a):=\sup a_{*}^{-1}((0,|a|])\leq\log|a|$.

######  Remark 5.1.

The above result may look contradictory with [19, Theorem 2.1], which reads
blow-up at arbitrarily large time, but what is actually proven in [19] is that
$\|u(t)\|_{L^{2}_{x}(\mathbb{R}/2\pi\mathbb{Z})}\sim\|u^{n}(t)\|_{\ell^{2}_{n}(\mathbb{N})}=\infty$
for $T_{0}:=\sum_{k\geq 2}\tfrac{1}{k^{2}}\log\tfrac{3k-3}{2k-3}\leq
t\leq\log\tfrac{|a|}{3}$, as well as exponential divergence of $u^{n}(t)$ for
$T_{0}\leq t<\log\tfrac{|a|}{3}$. The above result shows that the (first)
blow-up time is earlier (which is also obvious from the argument in [19]), and
uniformly bounded by $T_{**}$. Indeed, $3e^{t}>\tfrac{e^{t}}{1-e^{-2t}}$ for
$t>T_{1}:=\tfrac{1}{2}\log\tfrac{3}{2}$ and (5.17) imply $|a|>a_{*}(t)$ for
$T_{1}<t\leq\log\tfrac{|a|}{3}$, while $T_{1}<\tfrac{1}{4}\log 3<T_{0}$. Since
the inverse function of $\tau:(\tfrac{1}{2}\log 3,\infty)\ni
t\mapsto\tfrac{e^{t}}{1-e^{-2t}}$ satisfies $\tau^{-1}(a)=\log
a-a^{-2}+O(a^{-4})$ as $a\to\infty$, the time interval
$[T_{0},\log\tfrac{|a|}{3}]$ considered in [19] is inside the set where
$u(t)\not\in\mathscr{S}^{\prime}(\mathbb{R})$ with a uniform distance from the
boundary.

######  Remark 5.2.

The Cole-Hopf transform yields another solution $\tilde{u}$ by the formula

$\begin{split}v:=\sum_{n\in\mathbb{N}_{0}}\tfrac{a^{n}}{n!}e^{inx-n^{2}t},\quad\tilde{u}:=-v_{x}/v.\end{split}$
(5.21)

The blow-up occurs at $v(t,x)=0$, before which $\tilde{u}$ is analytic and the
same as the solution in $\mathcal{F}^{-1}\mathcal{X}$, namely
$\tilde{u}|_{0\leq t<T_{*}(a)}=u|_{0\leq t<T_{*}(a)}$.

To see what happens after that, let
$V(t,x):=\sum_{n\in\mathbb{N}_{0}}\tfrac{1}{n!}e^{inx-n^{2}t}$, which is
analytic on $H:=(0,\infty)\times\mathbb{C}$ satisfying the heat equation and
$v(t,x)=V(t,x-i\log a)$. Let $Z:=\\{(t,x)\in H\mid V(t,x)=0\\}$ and
$D:=\\{(t,x)\in Z\mid V_{x}(t,x)=0\\}$. Then [29, Theorem 4.1] implies that
$D$ is discrete in $H$. For any $(t_{0},x_{0})\in Z\setminus D$, the
Weierstrass preparation theorem yields analytic functions $z(t),A(t,x)$
defined around $(t,x)=(t_{0},x_{0})$ such that

$\begin{split}V(t,x)=A(t,x)(x-x_{0}-z(t)),\quad z(t_{0})=0,\quad
A(t_{0},x_{0})\not=0.\end{split}$ (5.22)

Let $E$ be the set of all $(t_{0},x_{0})\in Z\setminus D$ such that
$\mathop{\mathrm{Im}}z(t_{0}-\varepsilon)\not=0\leq\mathop{\mathrm{Im}}z(t_{0}-\varepsilon)\mathop{\mathrm{Im}}z(t_{0}+\varepsilon)$
for all small $\varepsilon>0$. Then the analyticity of $z$ implies existence
of a positive even integer $m$ such that
$\mathop{\mathrm{Im}}\partial_{t}^{k}z(t_{0})=0\not=\mathop{\mathrm{Im}}\partial_{t}^{m}z(t_{0})$
for $k<m$. Since $Z=\\{(t,x_{0}+z(t))\\}$ in a neighborhood of
$(t_{0},x_{0})$, we deduce that $E$ is also discrete in $H$. On the other
hand, $V(0,x)=e^{e^{ix}}$ has no zero for $x\in\mathbb{C}$, $V(t,x)\to 1$ as
$t\to\infty$ uniformly for bounded $\mathop{\mathrm{Im}}x$, and
$|V(t,x)-1|\leq e^{e^{-\mathop{\mathrm{Im}}x}}-1$. Hence for any
$\lambda\in\mathbb{R}$, there is $\delta>0$ such that for any $(t,x)\in Z$
with $\mathop{\mathrm{Im}}x>\lambda$ we have $\delta\leq t\leq 1/\delta$ and
$\mathop{\mathrm{Im}}x\leq 1/\delta$. Therefore,

$\begin{split}S_{\lambda}:=\\{(t,x)\in D\cup E\mid
0\leq\mathop{\mathrm{Re}}x\leq 2\pi,\
\mathop{\mathrm{Im}}x>\lambda\\}\end{split}$ (5.23)

is finite for any $\lambda\in\mathbb{R}$. Hence
$A:=\\{e^{-\mathop{\mathrm{Im}}x}\mid(t,x)\in S_{-\infty}\\}\subset(0,\infty)$
is locally finite.

Now let $a\in\mathbb{C}$ satisfy $|a|\geq a_{**}$ and $|a|\not\in A$. Then
$t_{0}:=T_{*}(a)\in(0,T_{**}]$ is the first $t>0$ such that $0\in
v(t,\mathbb{R})$. Let $\mathcal{B}(a):=\\{x\in\mathbb{R}\mid v(t_{0},x)=0\\}$
be the blow-up set, $x_{1}\in\mathcal{B}(a)$ and $x_{0}:=x_{1}-i\log a$. Then
$(t_{0},x_{0})\in Z\setminus(D\cup E)$. Let $z(t)$ as in (5.22). Since
$0\not\in v(t,\mathbb{R})$ for any $t\in(0,t_{0})$, we have
$\mathop{\mathrm{Im}}z(t)\not=0$ for $t<t_{0}$. Since $(t_{0},x_{0})\not\in
E$, the minimal integer $m$ such that
$\mathop{\mathrm{Im}}\partial_{t}^{m}z(t_{0})\not=0$ is odd. In other words,
the blow-up for $|a|\not\in A$ occurs when some zeros of $v$ crosses
$\mathbb{R}$ in $\mathbb{C}$. It also implies that $\mathcal{B}(a)$ is
discrete in $\mathbb{R}$. Let
$\mathcal{B}(a)\cap[0,2\pi)=\\{x_{1},\ldots,x_{J}\\}$ such that all $x_{j}$
are distinct. Then we have an open interval $I\ni t_{0}$ and analytic
functions $z_{j}(t)$ defined on $I$ for $j=1,\ldots,J$ such that
$v(t,x)=A_{j}(t,x)(x-x_{j}-z_{j}(t))$ around $(t_{0},x_{j})$, for some
analytic $A_{j}$ satisfying $A_{j}(t_{0},x_{j})\not=0$. Then there exists
$\\{c_{1},\ldots,c_{J}\\}\in\mathbb{C}\setminus\\{0\\}$ such that

$\begin{split}R(t,x):=\tilde{u}(t,x)-\sum_{j=1}^{J}c_{j}[e^{-i(x-x_{j}-z_{j}(t))}-1]^{-1}\end{split}$
(5.24)

is analytic for $(t,x)\in I\times\mathbb{R}$, and that each
$\mathop{\mathrm{Im}}z_{j}(t)$ changes its sign at $t=t_{0}$. The last summand
may be expanded by using

$\begin{split}[e^{-i(x-z)}-1]^{-1}=\begin{cases}\sum_{n\geq
1}e^{in(x-z)}&(\mathop{\mathrm{Im}}z<0),\\\ -\sum_{n\leq
0}e^{in(x-z)}&(\mathop{\mathrm{Im}}z>0).\end{cases}\end{split}$ (5.25)

Since $\tilde{u}$ is analytic for $t<t_{0}$ and
$\operatorname{supp}\mathcal{F}\tilde{u}\subset[1,\infty)$, we deduce that
$\mathop{\mathrm{Im}}z_{j}(t)<0$ for $t<t_{0}$ and
$\mathop{\mathrm{Im}}z_{j}(t)>0$ for $t>t_{0}$, for all $j$. Therefore

$\begin{split}\tilde{u}(t,x)=\begin{cases}\sum_{j=1}^{J}c_{j}\sum_{n\geq
1}e^{in(x-x_{j}-z_{j}(t))}+R(t,x)&(t_{0}-\varepsilon<t<t_{0})\\\
\sum_{j=1}^{J}c_{j}\sum_{n\leq
0}e^{in(x-x_{j}-z_{j}(t))}+R(t,x)&(t_{0}<t<t_{0}+\varepsilon).\end{cases}\end{split}$
(5.26)

In particular, $\tilde{u}(t)$ is discontinuous even in
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R})$ at the blow-up time
$t=t_{0}=T_{*}(a)$.

The above construction of blow-up works partly for the equation

$\begin{split}u_{t}=\Delta u+2u\cdot\nabla u,\quad
u(t,x):\mathbb{R}^{1+d}\to\mathbb{C}^{d},\end{split}$ (5.27)

with the general dimension $d\in\mathbb{N}$ and the initial data in the form

$\begin{split}u_{0}=-a\nabla\mathcal{F}^{-1}\mu,\quad a>0,\quad
0\leq\mu\in\mathcal{M}(\mathbb{R}^{d}),\quad\operatorname{supp}\mu\subset
B_{\mathbb{R}^{d}}(\mathrm{v},R),\end{split}$ (5.28)

for some $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$ and
$R\in(0,|\mathrm{v}|/\sqrt{2})$. Then the iteration

$\begin{split}&u(t)=-\nabla\sum_{n\in\mathbb{N}}a^{n}\mathcal{F}^{-1}U_{n}(t),\quad
U_{1}(t,\xi)=e^{-t|\xi|^{2}}\mu(\xi),\\\ &U_{n}(t,\xi)=\sum_{1\leq k\leq
n-1}\int_{0}^{t}e^{-(t-s)|\xi|^{2}}\int_{\mathbb{R}^{d}}(\xi-\eta)\cdot\eta
U_{k}(s,\xi-\eta)U_{n-k}(s,\eta)d\eta ds,\end{split}$ (5.29)

gives the unique global solution $u\in
C(\mathbb{R};\mathcal{F}^{-1}\mathcal{X}(\mathrm{v}))$ with
$\operatorname{supp}U_{n}\subset nB_{\mathbb{R}^{d}}(\mathrm{v},R)$. Let

$\begin{split}&\underline{b}:=\inf_{\xi\in\operatorname{supp}\mu}|\xi|\geq|\mathrm{v}|-R>0,\quad\bar{b}:=\sup_{\xi\in\operatorname{supp}\mu}|\xi|\in[\underline{b},|\mathrm{v}|+R],\\\
&\underline{c}:=\inf_{\xi,\eta\in\operatorname{supp}\mu}\xi\cdot\eta\geq\min(1-\tfrac{R^{2}}{|\mathrm{v}|^{2}},\tfrac{1}{2})(|\mathrm{v}|^{2}-2R^{2})>0,\\\
&\bar{c}:=\sup_{\xi,\eta\in\operatorname{supp}\mu}\xi\cdot\eta\in[\bar{c},\bar{b}^{2}].\end{split}$
(5.30)

Then the equation of $U_{n}$ is bounded from below and above by

$\begin{split}&\tfrac{1}{n}e^{-\bar{b}^{2}nt}\underline{f}_{n}(t)\mu^{*n}\leq
U_{n}(t)\leq\tfrac{1}{n}e^{-\underline{b}^{2}nt}\bar{f}_{n}(t)\mu^{*n},\quad\underline{f}_{1}(t):=\bar{f}_{1}(t):=1,\\\
&\quad\underline{f}_{n}(t):=\underline{c}\int_{0}^{t}ne^{-\bar{b}^{2}n(n-1)(t-s)}\sum_{1\leq
k\leq n-1}\underline{f}_{k}(s)\underline{f}_{n-k}(s)ds,\\\
&\quad\bar{f}_{n}(t):=\bar{c}\int_{0}^{t}ne^{-\underline{b}^{2}n(n-1)(t-s)}\sum_{1\leq
k\leq n-1}\bar{f}_{k}(s)\bar{f}_{n-k}(s)ds,\end{split}$ (5.31)

by induction on $n$. Then in the same way as before, we obtain inductively

$\begin{split}&(\bar{b}^{-2}\underline{c}(1-e^{-2\bar{b}^{2}t}))^{n-1}\leq\underline{f}_{n}(t)\leq\bar{f}_{n}(t)\leq(\underline{b}^{-2}\bar{c})^{n-1}.\end{split}$
(5.32)

The lower bound implies exponential growth of
$\|a^{n}U_{n}(t)\|_{\mathcal{M}}$ as $n\to\infty$ if

$\begin{split}a>\tfrac{3\sqrt{3}}{2}\bar{b}^{2}\underline{c}^{-1},\end{split}$
(5.33)

at some $t\in(0,\infty)$, where
$(\tfrac{3\sqrt{3}}{2})^{-1}=\max_{t>0}e^{-\bar{b}^{2}t}(1-e^{-2\bar{b}^{2}t})$
as before. By the Fourier support and positivity, it implies that
$u(t)\not\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ for those $t>0$. On the
other hand, the upper bound implies that $\|a^{n}U_{n}(t)\|_{\mathcal{M}}$
decays exponentially as $n\to\infty$ if

$\begin{split}0\leq a<\underline{b}^{2}\bar{c}^{-1},\end{split}$ (5.34)

uniformly in $t\geq 0$, and so $u$ is an analytic global solution on $t\geq
0$.

The above examples show that the global wellposedness without size restriction
as in this paper can not hold within the tempered distributions
$\mathscr{S}^{\prime}$, even with a smoothing linear part, a very mild
nonlinearity, and analytic initial data.

###  5.2. Solutions beyond the natural boundary

For the ODE $\dot{u}=u^{2}$, the solution formula (5.2) makes sense and smooth
except at most the single time $t=1/u_{0}$ for each $u_{0}\in\mathbb{C}$.
However, the global wellposedness in $\mathcal{F}^{-1}\mathcal{X}$ applies to
more exotic equations as well, for example

$\begin{split}\dot{u}=\mathscr{H}(u):=\sum_{n=0}^{\infty}u^{2^{n}}.\end{split}$
(5.35)

This $\mathscr{H}(u)$ is a well-known lacunary series with the natural
boundary $|u|=1$, so it may not be extended anywhere of $|u|\geq 1$ as a
holomorphic function. Nevertheless, the global wellposedness holds in
$\mathcal{F}^{-1}\mathcal{X}$, which includes initial data such as

$\begin{split}u_{0}(x)=ae^{ic\cdot x},\end{split}$ (5.36)

for any $a\in\mathbb{C}$ and $c\in\mathbb{R}^{d}_{>0}$. If we choose $|a|\geq
1$, then $\mathscr{H}(u_{0}(x))$ does not make sense for the holomorphic
function $\mathscr{H}$ at any $x\in\mathbb{R}^{d}$. On the other hand, the
local wellposedness holds for $\|u_{0}\|_{\mathcal{F}^{-1}\mathcal{M}}<1$ and
obtained by the iteration estimate as above, where the solutions are the same
as those in $\mathcal{F}^{-1}\mathcal{X}$. In this sense, the global solutions
in $\mathcal{F}^{-1}\mathcal{X}$ may be regarded as extension beyond the
analytic continuation on $\mathbb{C}$. Actually, we could also consider
everywhere divergent Taylor series if the initial Fourier support is uniformly
away from the boundary $\xi_{1}=0$, but we do not pursue it here.

##  6\. Optimality of the conditions for the solution space

In this section, we investigate the optimality of the function space
$\mathcal{X}$ and the stronger support condition for the global wellposedness
results, using the simple ODE (5.1) with the solution formula (5.2).

###  6.1. Frequency on the half space

The first example shows that $\hat{u}_{0}$ is not allowed to have point
measures either in negative frequency $\xi_{1}<0$ or on the boundary
$\xi_{1}=0$, for global existence in $\mathcal{F}^{-1}\mathscr{D}^{\prime}$.
This means that the half space is optimal for the Fourier support condition,
including the zero boundary condition.

Let $c\in\mathbb{R}^{d}$ and $u$ be the solution of (5.1) for the initial data

$\begin{split}u(0,x)=\cos(c\cdot
x),\quad\hat{u}(0,\xi)=\tfrac{1}{2}[\delta(\xi-c)+\delta(\xi+c)].\end{split}$
(6.1)

Note that for a single point measure $\delta(x-c)$ with $c\not=0$, we may
choose the direction $\mathrm{v}=c$ to satisfy
$\operatorname{supp}\hat{u}_{0}\subset\mathbb{R}^{d}_{>0}(\mathrm{v})$ and so
the global wellposedness follows.

Now we confirm blow-up of the above solution $u$ in
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$. Although it may look
obvious from the solution formula, the previous section suggests that such
very weak solutions might potentially survive beyond our intuition. The Cauchy
problem is locally wellposed in $\mathcal{F}^{-1}\mathcal{M}$, where we have

$\begin{split}u(t,x)=\tfrac{\cos(c\cdot x)}{1-t\cos(c\cdot x)}=\sum_{n\geq
0}t^{n}\cos^{n+1}(c\cdot x),\end{split}$ (6.2)

which is absolutely convergent in $\mathcal{F}^{-1}\mathcal{M}$ for $|t|<1$
and $u\in C((-1,1);\mathcal{F}^{-1}\mathcal{M})$. If $c\not=0$ then the
Fourier transform is in the form

$\begin{split}\hat{u}(t,\xi)=\sum_{j\in\mathbb{Z}}\sum_{n\geq
0}a_{j,n}t^{n}\delta(\xi-jc),\end{split}$ (6.3)

for certain non-negative coefficients $a_{j,n}$ that may be computed
explicitly. In particular at $\xi=0$, the binomial expansion of
$\cos^{n+1}(c\cdot x)=(\tfrac{e^{icx}+e^{-icx}}{2})^{n+1}$ yields

$\begin{split}\sum_{n\geq
0}a_{0,n}t^{n}=\sum_{k=1}^{\infty}\begin{pmatrix}2k\\\
k\end{pmatrix}\tfrac{t^{2k-1}}{2^{2k}}\geq\sum_{k=1}^{\infty}\tfrac{t^{2k-1}}{2k}\to\infty,\end{split}$
(6.4)

as $0<t\to 1-0$. The above inequality follows from
$\tfrac{a_{0,2k+1}}{a_{0,2k-1}}=\tfrac{2k+1}{2k+2}\geq\tfrac{k}{k+1}$ and
$a_{0,1}=\tfrac{1}{2}$. Hence, coupling $\hat{u}(t)$ with a test function
$\phi\in\mathscr{D}(\mathbb{R}^{d})$ with small support around $\xi=0$, we
deduce that $u(t)$ blows up in
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$ as $t\to 1-0$. In the
case of $c=0$, it is more direct from the formula
$\hat{u}(t,\xi)=\tfrac{1}{1-t}\delta(\xi)$.

###  6.2. Decay to the boundary

For finite measures on $\mathbb{R}^{d}_{\geq 0}$, the vanishing at the
boundary $\xi_{1}=0$ is equivalent to the decay as $\xi_{1}\to+0$. It is the
case for elements in $\mathcal{X}$, because they are required to be
$\mathbb{C}$-valued measures near $\xi_{1}=0$. Then it is natural to ask if
the latter condition is necessary. In other words, can we weaken or replace
the topology for the decay as $\xi_{1}\to+0$, or can we include $\hat{u}_{0}$
that is not a measure near $\xi_{1}=0$? The following shows that the measure
space is optimal at least among the family of Sobolev or Besov spaces (in the
Fourier side). Let

$\begin{split}B^{-0}:=\bigcap_{\sigma<0}(B^{\sigma}_{1,\infty}\cap
B^{\sigma}_{\infty,\infty})(\mathbb{R}^{d})\hookrightarrow\mathscr{D}^{\prime}(\mathbb{R}^{d})\end{split}$
(6.5)

be the Fréchet space endowed with a countable family of norms, e.g.,
$\|\varphi\|_{B^{-1/n}_{1,\infty}}+\|\varphi\|_{B^{-1/n}_{\infty,\infty}}$ or
$\|{\langle\nabla\rangle}^{-1/n}\varphi\|_{L^{1}\cap L^{\infty}}$ for
$n\in\mathbb{N}$. It is also a convolution algebra because

$\begin{split}\|f*g\|_{B^{2\sigma}_{p,\infty}}\lesssim\|f\|_{B^{\sigma}_{1,1}}\|g\|_{B^{\sigma}_{p,1}}\lesssim\|f\|_{B^{\sigma/2}_{1,\infty}}(\|g\|_{B^{\sigma/2}_{1,\infty}}+\|g\|_{B^{\sigma/2}_{\infty,\infty}}),\end{split}$
(6.6)

for any $\sigma<0$ and $p\in[1,\infty]$, but we can not solve (5.1) in
$\mathcal{F}^{-1}B^{-0}_{\geq 0}$. More precisely,

######  Proposition 6.1.

For every $d\in\mathbb{N}$, there is a sequence
$\\{f_{n}\\}_{n\in\mathbb{N}}\subset\mathscr{D}_{>0}$ convergent in $B^{-0}$
such that the global solution $u_{n}\in
C(\mathbb{R};\mathcal{F}^{-1}\mathcal{X})$ of (5.1) with
$\hat{u}_{n}(0)=f_{n}$ is unbounded in
$\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$ as $n\to\infty$ at any
fixed $t\not=0$. The iteration sequence $v_{n}$ defined by (5.3) with
$\hat{u}_{0}=\lim_{n\to\infty}f_{n}$ is also unbounded at any fixed $t\not=0$.

Note that the convergence from $\mathscr{D}_{>0}$ implies the decay of the
limit $\hat{u}_{0}$ to the boundary $\xi_{1}=0$ in the $B^{-0}$ norm. So we
may not include any negative Sobolev or Besov space up to the boundary
$\xi_{1}=0$, for the local wellposedness, or to construct the solution by
iteration.

###### Proof of Proposition 6.1.

For $k,n\in\mathbb{N}$, let

$\begin{split}&r_{k}:=1/k!,\quad N_{k}:=k^{k^{2}},\quad M_{k}:=2^{k^{2}},\\\
&g_{k}:=\chi(4r_{k+1}^{-1}|\xi-r_{k}e_{1}|)\exp(-iN_{k}\xi_{1}),\quad
f_{n}:=\sum_{k=1}^{n}M_{k}g_{k}.\end{split}$ (6.7)

Then $\operatorname{supp}g_{k}\subset\\{|\xi-r_{k}e_{1}|<r_{k+1}/2\\}$ implies
$g_{k},f_{n}\in\mathscr{D}_{>0}$, and we have

$\begin{split}&\check{g}_{k}:=\mathcal{F}^{-1}g_{k}=[r_{k+1}^{d}\check{\chi}(r_{k+1}x)e^{ir_{k}x_{1}}]|_{x\mapsto
x-N_{k}e_{1}},\end{split}$ (6.8)

where $\check{\chi}:=\mathcal{F}^{-1}\chi(4|\xi|)$, and

$\begin{split}\|{\langle\nabla\rangle}^{-\varepsilon}g_{k}\|_{(L^{1}\cap
L^{\infty})_{\xi}}\lesssim\|(1-\Delta)^{d}[{\langle
x\rangle}^{-\varepsilon}\check{g}_{k}](x+N_{k}e_{1})\|_{L^{1}_{x}}\lesssim
N_{k}^{-\varepsilon},\end{split}$ (6.9)

uniformly for $0\leq\varepsilon\leq 1$ and $k\in\mathbb{N}$. Since
$M_{k}/N_{k}^{\varepsilon}\to 0$ as $k\to\infty$ for any $\varepsilon>0$,
$f_{n}$ is convergent in $B^{-0}$.

The global solution $u_{n}$ is given by the formula
$\hat{u}_{n}(t)=\sum_{j=1}^{\infty}t^{j-1}f_{n}^{*j}$ as before, and the
support of $g_{1},\ldots,g_{n}$ implies that for any
$\varphi\in\mathscr{D}_{<r_{n-1}-r_{n}/2}$ we have

$\begin{split}\hat{u}_{n}(t)(\varphi)=\sum_{1\leq
j<s_{n}}t^{j-1}(M_{n}g_{n})^{*j}(\varphi)=\sum_{1\leq j\leq
n-1}t^{j-1}(M_{n}\check{g}_{n})^{j}(\hat{\varphi}),\end{split}$ (6.10)

where $s_{n}:=\tfrac{r_{n-1}-r_{n}/2}{r_{n}-r_{n+1}/2}=n-\tfrac{1}{2n+1}$.
Since $(r_{n}+\tfrac{1}{2}r_{n+1})(n-1)<r_{n-1}-\tfrac{1}{2}r_{n}$, we may
take $\varphi=\varphi_{n}:=(\bar{g}_{n})^{*(n-1)}$, then

$\begin{split}(2\pi)^{d}\hat{u}_{n}(t)(\varphi_{n})&=\sum_{1\leq j\leq
n-1}t^{j-1}M_{n}^{j}{\langle\check{g}_{n}^{j}|\check{g}_{n}^{n-1}\rangle}_{L^{2}(\mathbb{R}^{d})},\end{split}$
(6.11)

where the summand for $j=n-1$ is equal to

$\begin{split}t^{n-2}M_{n}^{n-1}\|r_{n+1}^{d}\check{\chi}(r_{n+1}x)\|_{L^{2(n-1)}}^{2(n-1)}=t^{n-2}M_{n}^{n-1}r_{n+1}^{d(2n-3)}\|\check{\chi}\|_{L^{2n-2}}^{2n-2},\end{split}$
(6.12)

while for $j<n-1$ it is bounded by

$\begin{split}t^{j-1}M_{n}^{j}r_{n+1}^{d(n+j-2)}\|\check{\chi}\|_{L^{n+j-1}}^{n+j-1}.\end{split}$
(6.13)

Since $\check{\chi}\in\mathscr{S}(\mathbb{R}^{d})$ with $\check{\chi}(0)>0$,
there are $c_{0},c_{1},c_{2},c_{3}\in(0,\infty)$ such that

$\begin{split}c_{0}c_{1}^{p}\leq\|\check{\chi}\|_{L^{p}}^{p}\leq
c_{2}c_{3}^{p}\end{split}$ (6.14)

for all $p\in[1,\infty)$. Since $M_{n}r_{n+1}^{d}=2^{n^{2}}((n+1)!)^{-d}\gg
A^{n}$ as $n\to\infty$ for any $A\in(1,\infty)$, we deduce that

$\begin{split}|\hat{u}_{n}(t)(\varphi_{n})|\gtrsim
t^{n-2}M_{n}^{n-1}r_{n+1}^{d(2n-3)}c_{1}^{2n-2}\geq
t^{-1}(c_{1}^{2}tM_{n}r_{n+1}^{2d})^{n-1}\end{split}$ (6.15)

for sufficiently large $n$ (depending on $d,t$).

On the other hand, since
$\operatorname{supp}\varphi_{n}\subset\\{|\xi|<2(n-1)r_{n}\\}\subset\\{|\xi|\leq
2\\}$ for all $n\in\mathbb{N}$, if $\hat{u}_{n}(t)$ is bounded in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$, then there exists $m\in\mathbb{N}$
such that for large $n$

$\begin{split}|\hat{u}_{n}(t)(\varphi_{n})|\leq
m\|\varphi_{n}\|_{\mathcal{B}^{m}}\leq
m\|g_{n}\|_{L^{1}}^{n-2}\|g_{n}\|_{\mathcal{B}^{m}}\leq
C(m)N_{n}^{m}.\end{split}$ (6.16)

Comparison of those two estimates as $n\to\infty$ leads to a contradiction,
since

$\begin{split}&\log(M_{n}r_{n+1}^{2d})=n^{2}\log 2-2d\log(n+1)!\sim n^{2},\\\
&\log(N_{n}^{m/(n-1)})=\tfrac{mn^{2}}{n-1}\log n\sim mn\log n.\end{split}$
(6.17)

Therefore, $\hat{u}_{n}(t)$ is not bounded in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$ at any $t\not=0$. The argument is
exactly the same for the iteration $v_{n}$ because of the formula (5.4). ∎

###  6.3. Integrability in the orthogonal directions

Next we consider if $\mathcal{X}$ needs the integrability condition in
$\xi^{\perp}$, which is also quite strong in some sense. For example in the
$L^{2}$-Sobolev spaces, we may easily observe that if $d\geq 2$ then

$\begin{split}\\{F\in
H^{s}(\mathbb{R}^{d})\mid\operatorname{supp}\hat{F}\subset\mathbb{R}^{d}_{\geq
0}\\}\subset\mathcal{F}^{-1}\mathcal{X}\iff s>\tfrac{d-1}{2}.\end{split}$
(6.18)

On the other hand, if $\mathcal{R}\cap\mathbb{R}^{d}_{<l}$ is bounded for each
$l>0$, then we have $(\mathcal{F}H^{s})_{\mathcal{R}}\subset\mathcal{X}$ for
all $s\in\mathbb{R}$.

First, the necessity of (lower) bounded regularity on $\mathbb{R}^{d}_{<l}$
for each $l>0$ is observed by the simple example

$\begin{split}\mathcal{X}\ni f_{N}:=\sum_{n=1}^{N}c_{n}[\delta(\xi-
le_{1}-ne_{2})+(-\partial_{1})^{n}\delta(\xi-le_{1}+ne_{2})]\to
f_{\infty}\text{ in }\mathscr{D}^{\prime}_{\geq 0},\end{split}$ (6.19)

as $N\to\infty$, where $l>0$ and
$\\{c_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ are arbitrarily fixed. For any
$\varphi\in\mathscr{D}(\mathbb{R}^{d})$ with
$\operatorname{supp}\varphi\subset B_{\mathbb{R}^{d}}(2le_{1},1)$, we have

$\begin{split}(f_{N}*f_{N})(\varphi)=2\sum_{n=1}^{N}c_{n}^{2}(\partial_{1}^{n}\varphi)(2le_{1}),\end{split}$
(6.20)

so $f_{N}*f_{N}$ is not bounded in $\mathscr{D}^{\prime}(\mathbb{R}^{d})$ as
$N\to\infty$, no matter how fast $c_{n}\to 0$ as $n\to\infty$. Since
$\operatorname{supp}f_{N}^{*j}\subset\\{\xi_{1}=jl\\}$, it is easy to deduce
that the solution $u_{N}$ of (5.1) with $\hat{u}_{N}(0)=f_{N}$ is also
unbounded in $\mathscr{D}^{\prime}(\mathbb{R}^{d})$ at any $t\not=0$, as well
as for the iteration sequence for the initial data $\hat{u}(0)=f_{\infty}$.

Second, the optimality of $L^{1}_{\xi}$ may be observed from the example

$\begin{split}\mathcal{D}_{>0}\ni
g_{N}:=\chi(\xi_{1}-3){\langle\xi^{\perp}\rangle}^{-\sigma}\chi(|\xi|/N)\to
g_{\infty}:=\chi(\xi_{1}-3){\langle\xi^{\perp}\rangle}^{-\sigma}\text{ in
}\mathcal{B}^{\infty}\end{split}$ (6.21)

as $N\to\infty$, where $\sigma\in(0,d-1)$ is arbitrarily chosen. Note that
$\operatorname{supp}g_{N},\operatorname{supp}g_{\infty}\subset\mathbb{R}^{d}_{(1,5)}$.
For $n\in\mathbb{N}$ satisfying $\tfrac{n-1}{n}<\tfrac{\sigma}{d-1}$, we have

$\begin{split}g_{N}^{*n}\to
g_{\infty}^{*n}=\chi^{*n}(\xi_{1}-3n)({\langle\xi^{\perp}\rangle}^{-\sigma})^{*n}\text{
in }\mathscr{D}^{\prime}_{\geq
0},\quad({\langle\xi^{\perp}\rangle}^{-\sigma})^{*n}\sim{\langle\xi^{\perp}\rangle}^{(n-1)(d-1)-n\sigma},\end{split}$
(6.22)

but if $\tfrac{n-1}{n}>\tfrac{\sigma}{d-1}$ then $g_{N}^{*n}$ is unbounded as
$N\to\infty$ in $L^{1}_{loc}$ and also in $\mathscr{D}^{\prime}$.

Thus in the same way as before, we observe the unboundedness of solutions for
$\hat{u}(0)=g_{N}$ and for the iteration sequence of $\hat{u}(0)=g_{\infty}$.
This means that the condition $X^{l}\hat{u}_{0}\in W^{-m,1}(\mathbb{R}^{d})$
may not be replaced with $W^{-m,p}(\mathbb{R}^{d})$, for any $p>1$, in order
to keep the local wellposedness of (5.1). Note also that
$\mathcal{F}^{-1}g_{N}\to\mathcal{F}^{-1}g_{\infty}$ converges in
$H^{s}(\mathbb{R}^{d})$ for $s<\sigma-\tfrac{d-1}{2}$. Hence $u_{0}\in
H^{s}(\mathbb{R}^{d})$ is not enough for $s<\tfrac{d-1}{2}$, even if the
Fourier support is in a bounded interval of $\xi_{1}$ away from $0$. However,
the above example does not preclude the endpoint case $s=\tfrac{d-1}{2}$, for
which the iterated convolution grows only logarithmically for
$|\xi^{\perp}|\to\infty$.

###  6.4. Cone support for unbounded coefficients

For $d\geq 2$ and general smooth coefficients $\hat{L}$, the support
restriction to a cone region is necessary, as the following example shows.
Here we include a non-negative linear part, namely

$\begin{split}u_{t}=Lu+u^{2},\quad 0\leq\hat{L}(\xi)\in
C^{\infty}(\mathbb{R}^{d}).\end{split}$ (6.23)

Suppose that $\hat{u}_{0}\in L^{1}(\mathbb{R}^{d})$, supported and continuous
on $\\{|\xi^{\perp}|\leq R_{0}(\xi_{1})\\}$, and positive on the interior, for
some continuous $R_{0}:[0,\infty)\to[0,\infty)$ satisfying $R_{0}(0)=0$.

Suppose for contradiction that $R_{0}(a)/a\to+\infty$ as $a\to+0$ but $u$ is
global for $t\geq 0$. Then the positivity of the data and the coefficients
implies

$\begin{split}\begin{cases}\hat{u}(t)\geq\hat{u}_{0},\\\
\hat{u}\geq\hat{u}_{1}\implies\hat{u}(t)\geq\int_{0}^{t}\hat{u}_{1}(s)*\hat{u}_{1}(s)ds.\end{cases}\end{split}$
(6.24)

Iteration of those inequalities yields a lower bound

$\begin{split}t>0,\
\xi_{1}>0\implies\hat{u}(t,\xi)\geq\hat{u}_{*}(t,\xi)>0,\end{split}$ (6.25)

for some continuous $\hat{u}_{*}$ independent of $\hat{L}$. $\hat{u}_{*}$ is
positive everywhere on $\xi_{1}>0$ because of $R_{0}(a)/a\to\infty$ and
$\hat{u}_{0}>0$ on $|\xi|<R_{0}(\xi_{1})$. Then we may choose $\hat{L}(\xi)$
such that

$\begin{split}1\leq\xi_{1}\leq 2\implies
e^{\hat{L}(\xi)}\hat{u}_{*}(1,\xi)\geq 1.\end{split}$ (6.26)

Then $\hat{u}(2,\xi)\geq 1$ on $\mathbb{R}^{d}_{[1,2]}$. Injecting it into
(6.24) leads to blow-up of $\hat{u}$ for $t>2$ because $1$ is not integrable
for $\xi_{2}\in\mathbb{R}$.

So we need $\limsup_{a\to+0}R_{0}(a)/a<\infty$ to avoid such blow-up. Then
Lemma 4.4 yields $\mathcal{R}=\\{|\xi|\leq
R(\xi_{1})\\}\supset\operatorname{supp}\hat{u}_{0}$ so that the global
wellposedness holds in $\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}$.

##  7\. Global solutions with locally integrable Fourier transform

In this section, we consider solutions in
$\mathcal{F}^{-1}L^{1}_{loc}(\mathbb{R}^{d})\subset\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R}^{d})$,
with the Fourier support in the half space, assuming that $\hat{L}$ decays on
the boundary. This allows considerable simplification of both the function
spaces and the construction of unique global solutions. Note that for any
$\nu\in C_{0}^{\scaleto{\nearrow}{4pt}}$, the closure of $\mathscr{D}_{>0}$ in
$\mathcal{X}^{0,\nu}$ is the weighted $L^{1}$ space

$\begin{split}\mathcal{X}^{0,\nu}_{0}:=\\{F\in
L^{1}_{loc}(\mathbb{R}^{d})\mid\operatorname{supp}F\subset\mathbb{R}^{d}_{\geq
0},\ \nu(\xi_{1})^{-1}F\in L^{1}(\mathbb{R}^{d})\\}.\end{split}$ (7.1)

The solutions in this section belong to such spaces (upto rotation in $\xi$).
Note that $u_{0}\in\mathcal{F}^{-1}L^{1}_{loc}(\mathbb{R}^{d})$ is decaying at
spatial infinity in the average sense (4.26).

Let $\mathcal{R}\subset\mathbb{R}^{d}$ be a closed subset satisfying
$\mathcal{R}+\mathcal{R}\subset\mathcal{R}$. Let
$\mathfrak{p}:\mathcal{R}\to[0,\infty)$ be continuous and super-additive,
namely $\mathfrak{p}(\xi+\eta)\geq\mathfrak{p}(\xi)+\mathfrak{p}(\eta)$ for
all $\xi,\eta\in\mathcal{R}$. Let $\hat{L},\hat{M},\hat{N}:\mathcal{R}\to
M(n_{1}),M(n_{2},n_{1}),M(n_{1},n_{3})$ be measurable, satisfying for some
$0<c_{0}\leq 1\leq C_{0}<\infty$

$\begin{split}\sigma_{\max}(\tfrac{\hat{L}+\hat{L}^{*}}{2})\leq(1-c_{0})\mathfrak{p},\quad\|\hat{N}\|_{M(n_{1})}+\|\hat{M}\|_{M(n_{2},n_{1})}\leq
C_{0}{\langle\mathfrak{p}\rangle},\end{split}$ (7.2)

almost everywhere on $\mathcal{R}\subset\mathbb{R}^{d}$, where $\sigma_{\max}$
denotes the maximal eigenvalue of the self-adjoint matrix. The above condition
of $\hat{L}$ implies $\|e^{t\hat{L}(\xi)}\|_{M(n_{1})}\leq
e^{t(1-c_{0})\mathfrak{p}(\xi)}$ a.e. on $\mathcal{R}$. Hence, all of
$e^{t\hat{L}},\hat{N},\hat{M}$ are locally bounded on $\mathcal{R}$.

The continuity and super-additivity of $\mathfrak{p}$ may be implemented as
follows. If (7.2) is satisfied by a locally bounded
$\mathfrak{p}_{0}:\mathcal{R}\to[0,\infty)$ in place of $\mathfrak{p}$, such
that

$\begin{split}\sup\\{|\xi|^{-1}\mathfrak{p}_{0}(\xi)\mid\xi\in\mathcal{R},\
|\xi|<\delta\\}<\infty\end{split}$ (7.3)

for some $\delta>0$, then Lemma 4.4 yields a continuous and super-additive
$\mathfrak{p}:\mathcal{R}\to[0,\infty)$ satisfying
$\mathfrak{p}_{0}\leq\mathfrak{p}$, hence (7.2) holds for this $\mathfrak{p}$.

A difference from the previous setting in $\mathcal{X}$ is that
$\sigma_{\max}(\tfrac{\hat{L}+\hat{L}^{*}}{2})$ is required to decay as
$\mathfrak{p}\to 0$, prohibiting uniform instability of $L$ near $\xi_{1}=0$
for $\mathcal{R}=\mathbb{R}^{d}_{\geq 0}$. It is actually possible to allow
$\hat{L}$ to be merely bounded as $\mathfrak{p}\to 0$ as before, but it will
complicate the weight for the solution space (double-exponential in time).
Since many physical examples satisfy the decay condition, we choose the
simplicity in this section.

The nonlinearity $\mathscr{H}:\mathbb{C}^{n_{2}}\to\mathbb{C}^{n_{3}}$ is
assumed to be holomorphic around $0\in\mathbb{C}^{n_{2}}$ and vanishing to the
order $\mathscr{H}(z)=O(z^{k_{0}+1})$ for some $k_{0}\in\mathbb{N}$, namely

$\begin{split}\mathscr{H}(z)=\sum_{\alpha\in\mathbb{N}_{0}^{n_{2}},\
|\alpha|\geq
k_{0}+1}\tfrac{1}{\alpha!}\mathscr{H}^{(\alpha)}(0)z^{\alpha}\end{split}$
(7.4)

is absolutely convergent in a $0$-neighborhood. As $k_{0}\geq 1$ increases, we
may take the bigger space of initial data. In the previous setting in
$\mathcal{X}$, it was fixed to $k_{0}=1$. Under those assumptions on
$L,M,N,\mathscr{H}$, we consider the equation as before, namely

$\begin{split}u=e^{tL(D)}u_{0}+\int_{0}^{t}e^{(t-\tau)L(D)}N(D)\mathscr{H}(M(D)u(\tau))d\tau.\end{split}$
(7.5)

For the initial data, we introduce a class of weighted $L^{1}$ spaces. For any
$k_{0}>0>s$, $\mathcal{R}\subset\mathbb{R}^{d}$ and
$\mathfrak{p}:\mathcal{R}\to[0,\infty)$, let
$\mathcal{Z}_{\mathfrak{p}}^{k_{0},s}(\mathcal{R})$ be the set of all Banach
spaces $Z\hookrightarrow L^{1}_{loc}(\mathbb{R}^{d})$ satisfying for all
$\varphi\in Z$

  1. (1)

$\operatorname{supp}\varphi\subset\mathcal{R}$ and
$\||\varphi|\|_{Z}=\|\varphi\|_{Z}$.

  2. (2)

$\sup_{c\in\mathcal{R}}\|\varphi(\xi-c)\|_{Z}\leq\|\varphi\|_{Z}$.

  3. (3)

$\|W_{s,k_{0}}(\mathfrak{p})\varphi\|_{L^{1}(\mathcal{R})}\leq\|\varphi\|_{Z}$,

where the decreasing function $W_{s,k_{0}}:(0,\infty)\to(0,\infty)$ is defined
by

$\begin{split}W_{s,k_{0}}(\rho):=\max(\rho^{-1/k_{0}},\rho-1/s)e^{s\rho}.\end{split}$
(7.6)

$W_{s,k_{0}}(\mathfrak{p})$ is decreasing for $\mathfrak{p},-s,k_{0}>0$.

A typical setting is the following. Let $P:[0,\infty)\to[0,\infty)$ be
continuous and super-additive. Then a maximal element of
$\mathcal{Z}^{k_{0},s}_{P(\xi_{1})}(\mathbb{R}^{d}_{\geq 0})$ is the weighted
$L^{1}$ on the half space:

$\begin{split}&L^{k_{0},s}_{P}:=\\{\varphi\in
L^{1}_{loc}(\mathbb{R}^{d})\mid\operatorname{supp}\varphi\subset\mathbb{R}^{d}_{\geq
0},\ \|\varphi\|_{L^{k_{0},s}_{P}}<\infty\\},\\\
&\quad\|\varphi\|_{L^{k_{0},s}_{P}}:=\|W_{s,k_{0}}(P(\xi_{1}))\varphi\|_{L^{1}(\mathbb{R}^{d}_{\geq
0})},\end{split}$ (7.7)

for which the property (2) follows from the monotonicity of $W_{s,k_{0}}$.
Moreover,

$\begin{split}1/W_{s,k_{0}}(P)\in C_{0}^{\scaleto{\nearrow}{4pt}},\quad
L^{k_{0},s}_{P}=\mathcal{X}^{0,1/W_{s,k_{0}}(P)}\end{split}$ (7.8)

with equivalent norms. Conversely, for any $\nu\in
C^{\scaleto{\nearrow}{4pt}}_{0}$ satisfying

$\begin{split}\limsup_{t\to+0}t^{-1/k_{0}}\nu(t)<\infty,\end{split}$ (7.9)

let $P(t):=t\sup_{0<\tau<t}\nu(\tau)^{k_{0}}/\tau$. Then $P$ is continuous and
super-additive, satisfying for any $s<0$, $\nu\leq P^{1/k_{0}}\leq
C_{s,k_{0}}/W_{s,k_{0}}(P)$ with some constant $C_{s,k_{0}}\in(0,\infty)$
depending only on $s<0<k_{0}$. Hence $\mathcal{X}^{0,\nu}_{0}\hookrightarrow
L^{k_{0},s}_{P}$.

We may also consider anisotropic $L^{p}$ spaces. Let
$d_{0}\in\\{1,\ldots,d\\}$, $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$,
$\lambda>0$,

$\begin{split}P_{\mathrm{v}}(\xi):=\mathrm{v}\cdot\xi,\quad\mathcal{R}:=\\{\xi\in\mathbb{R}^{d}_{\geq
0}(\mathrm{v}):|\bar{\xi}|\leq\lambda P_{\mathrm{v}}(\xi)\\},\end{split}$
(7.10)

where we denote
$\xi=(\bar{\xi},\tilde{\xi})\in\mathbb{R}^{d_{0}}\times\mathbb{R}^{d-d_{0}}$
for any $\xi\in\mathbb{R}^{d}$. Then

$\begin{split}&Z:=\\{\varphi\in
L^{1}_{loc}(\mathbb{R}^{d})\mid\operatorname{supp}\varphi\subset\mathcal{R},\
\|\varphi\|_{Z}:=\|(P_{\mathrm{v}}^{-\sigma}+1)\varphi\|_{L^{r}_{\bar{\xi}}L^{1}_{\tilde{\xi}}}/C<\infty\\},\end{split}$
(7.11)

belongs to $\mathcal{Z}^{k_{0},s}_{P_{\mathrm{v}}}(\mathcal{R})$ for any $s<0$
for some $C\in(0,\infty)$, provided that
$\tfrac{d_{0}}{r^{\prime}}+\sigma>\tfrac{1}{k_{0}}$. Indeed, the property (2)
follows from the translation invariance of $L^{r}L^{1}$ together with
monotonicity of $P_{\mathrm{v}}$ on $\mathbb{R}^{d}_{\geq 0}(\mathrm{v})$,
while (3) follows by Hölder:

$\begin{split}&\|\max(P_{\mathrm{v}}^{-1/k_{0}},P_{\mathrm{v}})e^{sP_{\mathrm{v}}}\varphi\|_{L^{1}_{\xi}(\mathcal{R})}\lesssim\|1_{|\bar{\xi}|<1}|\bar{\xi}|^{\sigma-1/k_{0}}+|\bar{\xi}|e^{s|\bar{\xi}|}\|_{L^{r^{\prime}}_{\bar{\xi}}}\|(P_{\mathrm{v}}^{-\sigma}+1)\varphi\|_{L^{r}_{\bar{\xi}}L^{1}_{\tilde{\xi}}}.\end{split}$
(7.12)

The following is the unique global existence in those spaces.

######  Theorem 7.1.

Let $d,n_{1},n_{2},n_{3},k_{0}\in\mathbb{N}$ and let
$\mathcal{R},\mathfrak{p},L,M,N,\mathscr{H}$ be as above. Then for any $s<0$,
$a\geq 2$ and $B>0$, there exists $b\in[1,\infty)$ such that for any
$Z\in\mathcal{Z}_{\mathfrak{p}}^{k_{0},s}(\mathcal{R})$ and any
$u_{0}\in\mathcal{F}^{-1}L^{1}_{loc}(\mathbb{R}^{d})^{n_{1}}$ satisfying
$\operatorname{supp}\hat{u}_{0}\subset\mathcal{R}$ and
$\|e^{s\mathfrak{p}^{a}}\hat{u}_{0}\|_{Z}\leq B$, there exists a unique global
solution $u$ of (7.5) satisfying

$\begin{split}e^{b(s\mathfrak{p}^{a}-t\mathfrak{p})}\hat{u}(t)\in[Z(L^{\infty}_{t}(0,\infty))\cap
C([0,\infty);Z)]^{n_{1}}.\end{split}$ (7.13)

###### Proof.

We show that the equation (7.5) defines a contraction mapping on closed balls
in the Banach space of measurable
$\hat{u}:[0,\infty)\times\mathcal{R}\to\mathbb{C}^{n_{1}}$ with the norm

$\begin{split}\|u\|_{Z^{b,s}_{\mathfrak{p}}}:=\|e^{b(s\mathfrak{p}^{a}-t\mathfrak{p})}\hat{u}(t)\|_{Z(L^{\infty}_{t}(0,\infty))},\end{split}$
(7.14)

if $b\geq 1$ is chosen large enough. For the initial data part,
$\sigma_{\max}(\tfrac{1}{2}(\hat{L}+\hat{L}^{*}))\leq\mathfrak{p}$ on
$\mathcal{R}\supset\operatorname{supp}\hat{u}_{0}$ implies

$\begin{split}\|\mathcal{F}e^{tL}u_{0}\|_{Z^{b,s}_{\mathfrak{p}}}\leq\|e^{t\hat{L}}\hat{u}_{0}\|_{Z^{1,s}_{\mathfrak{p}}}\leq\|e^{s\mathfrak{p}^{a}}\hat{u}_{0}\|_{Z}\leq
B.\end{split}$ (7.15)

For the nonlinear Duhamel part, it suffices to estimate for $k\geq k_{0}$

$\begin{split}U(t,\xi)&:=e^{b(s\mathfrak{p}^{a}-t\mathfrak{p})}\int_{0}^{t}|e^{(t-\tau)\hat{L}}{\langle\mathfrak{p}\rangle}({\langle\mathfrak{p}\rangle}\hat{u}^{0})*\cdots*({\langle\mathfrak{p}\rangle}\hat{u}^{k})(\tau)|d\tau\\\
&\leq\int_{0}^{t}e^{(\tau-t)c_{0}b\mathfrak{p}}\mathfrak{p}G(\xi)d\tau=\tfrac{1-e^{-tc_{0}b\mathfrak{p}}}{c_{0}b}G(\xi),\end{split}$
(7.16)

in terms of
$U_{j}(\xi):=\|e^{b(s\mathfrak{p}^{a}-t\mathfrak{p})}\hat{u}^{j}\|_{L^{\infty}_{t}(0,\infty)}$,
where

$\begin{split}&G(\xi):=\int_{\xi=\xi^{0}+\cdots+\xi^{k}}\tfrac{{\langle\mathfrak{p}\rangle}}{\mathfrak{p}}e^{bs\Omega}\prod_{0\leq
j\leq k}{\langle\mathfrak{p}(\xi^{j})\rangle}U_{j}(\xi^{j})d\xi^{1}\cdots
d\xi^{k},\\\ &\quad\Omega:=\mathfrak{p}(\xi)^{a}-\sum_{0\leq j\leq
k}\mathfrak{p}(\xi^{j})^{a}.\end{split}$ (7.17)

The above estimate by $G$ used the super-additivity of $\mathfrak{p}$ and the
assumption on $\hat{L}$ in the form $\|e^{(t-\tau)\hat{L}}\|_{M(n_{1})}\leq
e^{(t-\tau)(1-c_{0})\mathfrak{p}}$ and
$\mathfrak{p}(\xi)\geq\mathfrak{p}(\xi^{0})+\cdots+\mathfrak{p}(\xi^{k})$.

In estimating $G$, by symmetry, we may assume that
$\mathfrak{p}(\xi^{0})=\max_{0\leq j\leq k}\mathfrak{p}(\xi^{j})$. We split
this domain of $\xi^{0},\ldots,\xi^{k}$ into three regions, depending on a
parameter $\theta\in(0,1]$. The super-additivity of $\mathfrak{p}$, the
binomial expansion and Young imply

$\begin{split}&\Omega\geq\mathfrak{p}(\xi)^{a}-\mathfrak{p}(\xi^{0})^{a}-\mathfrak{p}(\xi-\xi^{0})^{a}\geq
a\mathfrak{p}(\xi^{0})^{a-1}\mathfrak{p}(\xi-\xi^{0}),\\\
&\mathfrak{p}(\xi)\geq\mathfrak{p}(\xi^{1})+\cdots+\mathfrak{p}(\xi^{k})\geq
k[\mathfrak{p}(\xi^{1})\cdots\mathfrak{p}(\xi^{k})]^{1/k}.\end{split}$ (7.18)

If $\mathfrak{p}(\xi^{0})\leq\theta$ then $\mathfrak{p}(\xi^{j})\leq\theta$
for all $j$, so we just discard $\Omega\geq 0$ to get

$\begin{split}G|_{\mathfrak{p}(\xi^{0})\leq\theta}\leq{\langle\theta\rangle}^{k+2}U_{0}*V_{1}^{0}*\cdots*V_{k}^{0},\quad
V_{j}^{0}:=\mathfrak{p}^{-1/k}U_{j}|_{\mathfrak{p}\leq\theta}.\end{split}$
(7.19)

If $\mathfrak{p}(\xi^{0})\geq\theta$ and
$\mathfrak{p}(\xi-\xi^{0})\geq\theta$, then we have
$\mathfrak{p}(\xi)\geq\theta$ and

$\begin{split}&\Omega\geq(a-1)\theta\mathfrak{p}(\xi^{0})^{a-1}+\theta^{a-1}\mathfrak{p}(\xi-\xi^{0})\geq\theta^{a-1}[\mathfrak{p}(\xi^{0})+\cdots+\mathfrak{p}(\xi^{k})],\\\
&\quad
G|_{\mathfrak{p}(\xi^{0}),\mathfrak{p}(\xi-\xi^{0})\geq\theta}\lesssim\tfrac{{\langle\theta\rangle}}{\theta}\tfrac{{\langle\sigma\rangle}}{|\sigma|}U_{0}*V_{1}^{1}*\cdots*V_{k}^{1},\quad
V_{j}^{1}:={\langle\mathfrak{p}\rangle}e^{\sigma\mathfrak{p}}U_{j},\end{split}$
(7.20)

where $\sigma:=b\theta^{a-1}s<0$ and the $\xi^{0}$ dependent weight is bounded
by
${\langle\rho\rangle}e^{\sigma\rho}\lesssim|\sigma|^{-1}{\langle\sigma\rangle}$.
If $\mathfrak{p}(\xi^{0})\geq\theta\geq\mathfrak{p}(\xi-\xi^{0})$, then
$\mathfrak{p}(\xi^{j})\leq\theta\leq\mathfrak{p}(\xi)$ for $j=1,\ldots,k$, and

$\begin{split}&\Omega\geq
a\mathfrak{p}(\xi^{0})^{a-1}k[\mathfrak{p}(\xi^{1})\cdots\mathfrak{p}(\xi^{k})]^{1/k},\quad
e^{bs\Omega}\leq(b|s|\Omega)^{-1},\\\ &\quad
G|_{\mathfrak{p}(\xi^{0})\geq\theta\geq\mathfrak{p}(\xi-\xi^{0})}\leq\tfrac{{\langle\theta\rangle}^{2}}{\theta^{2}}\tfrac{\theta}{ak|\sigma|}U_{0}*V_{1}^{0}*\cdots*V_{k}^{0}.\end{split}$
(7.21)

Thus we obtain for $k\geq k_{0}$

$\begin{split}&G|_{\mathfrak{p}(\xi^{0})=\max_{j}\mathfrak{p}(\xi^{j})}\lesssim\tfrac{{\langle\theta\rangle}^{k+2}}{\theta}\tfrac{{\langle\sigma\rangle}}{|\sigma|}U_{0}*V_{1}^{2}*\cdots*V_{k}^{2},\quad
V_{j}^{2}:=V_{j}^{0}+V_{j}^{1}.\end{split}$ (7.22)

Since all $U_{j},V_{j}$ are supported on $\mathcal{R}$, and
$Z\in\mathcal{Z}^{k_{0},s}_{\mathfrak{p}}$, the property (2) of
$\mathcal{Z}^{k_{0},s}_{\mathfrak{p}}$ implies

$\begin{split}\|U_{0}*V_{1}^{2}*\cdots*V_{k}^{2}\|_{Z}\leq\|U_{0}\|_{Z}\prod_{1\leq
j\leq k}\|V_{j}^{2}\|_{L^{1}},\end{split}$ (7.23)

while the property (3) implies for $\sigma\leq s$ that
$\|V_{j}^{2}\|_{L^{1}}\leq\kappa\|U_{j}\|_{Z}$ with

$\begin{split}\kappa:=\sup_{\rho>0}\tfrac{\rho^{-1/k}|_{\rho\leq\theta}+{\langle\rho\rangle}e^{\sigma\rho}}{\max(\rho^{-1/k_{0}},\rho-1/s)e^{s\rho}}\leq\theta^{-1/k+1/k_{0}}e^{-s\theta}+{\langle
s\rangle}e^{(\sigma-s)\theta}.\end{split}$ (7.24)

For any $k_{0}\in\mathbb{N}$, $s<0$ and $\varepsilon>0$, there are
$\theta\in(0,1)$ and $b\in[1,\infty)$ such that $\kappa\leq\varepsilon$ for
all $k\geq k_{0}+1$. For $k=k_{0}$, we choose $\theta=1$. Then

$\begin{split}\|U_{0}*V_{1}^{2}*\cdots*V_{k}^{2}\|_{Z}\leq\begin{cases}(2e^{-s})^{k_{0}}\prod_{0\leq
j\leq k_{0}}\|U_{j}\|_{Z}&(k=k_{0})\\\ \varepsilon^{k}\prod_{0\leq j\leq
k}\|U_{j}\|_{Z}&(k\geq k_{0}+1).\end{cases}\end{split}$ (7.25)

Injecting it into (7.16), (LABEL:def_G) and (LABEL:est_G-conv), we may exploit
the denominator $b$ in (7.16), taking $b\geq 1$ larger if necessary. Thus we
obtain

######  Lemma 7.2.

For any $-\infty<s<0<c_{0},\varepsilon<1$, $a\geq 2$ and $k_{0}\in\mathbb{N}$,
there exists $b\in[1,\infty)$ such that for any $\mathcal{R},\mathfrak{p},L$
as above, any $Z\in\mathcal{Z}^{k_{0},s}_{\mathfrak{p}}(\mathcal{R})$, any
$k\in\mathbb{N}$ with $k\geq k_{0}$ and any $u^{0},\ldots,u^{k}$ satisfying
$\operatorname{supp}\hat{u}^{j}(t)\subset\mathcal{R}$ for all $t\geq 0$, we
have

$\begin{split}\|U\|_{Z(L^{\infty}_{t}(0,\infty))}\leq\varepsilon^{k}\prod_{0\leq
j\leq k}\|\hat{u}^{j}\|_{Z(L^{\infty}_{t}(0,\infty))},\end{split}$ (7.26)

for $U$ defined by (7.16).

The above lemma yields an estimate on the Duhamel nonlinear part

$\begin{split}\|\int_{0}^{t}e^{(t-\tau)L}N(D)\mathscr{H}(M(D)u)d\tau\|_{Z^{b,s}_{\mathfrak{p}}}\leq\sum_{|\alpha|\geq
k_{0}}C_{0}\tfrac{\|\partial^{\alpha}\mathscr{H}(0)\|_{\mathbb{C}^{n_{3}}}}{\alpha!}(C_{0}\varepsilon\|u\|_{Z^{b,s}_{\mathfrak{p}}})^{|\alpha|},\end{split}$
(7.27)

and a similar estimate for the difference. Hence, if we choose
$\varepsilon\in(0,1)$ small enough depending on $B,\mathscr{H},C_{0}$, then
the map $u\mapsto
e^{tL}u_{0}+\int_{0}^{t}e^{(t-\tau)L}\mathscr{H}(u(\tau))d\tau$ is a
contraction on the closed ball $\\{u\in
Z^{b,s}_{\mathfrak{p}}\mid\|u\|_{Z^{b,s}_{\mathfrak{p}}}\leq B\\}$, and the
unique global solution of (7.5) is obtained as its unique fixed point.

Since the Duhamel formula is continuous in $t$ at each $\xi$, $u\in
C([0,\infty);Z)$ follows if $Z$ satisfies the dominated convergence.
Otherwise, it suffices to take $b$ slightly bigger. ∎

##  8\. Examples of the equations

We apply the above abstract results to some typical nonlinear evolution
equations which come from physics, including the Navier-Stokes, the Euler, the
nonlinear Klein-Gordon equations, as well as some nonlinear and nonlocal
equations. The application is straightforward, leading to new global
wellposedness and unique existence results for more wild initial data than the
standard settings.

###  8.1. Incompressible Navier-Stokes and Euler equations

Consider the Cauchy problem for the incompressible Navier-Stokes and Euler
equations:

$\displaystyle
u_{t}-\varepsilon_{1}\partial_{1}^{2}u-\varepsilon_{\perp}\Delta_{\perp}u+u\cdot\nabla
u+\nabla p=0,\ \ {\rm div}\,u=0,$ (8.1)

where $u(t,x):\mathbb{R}^{1+d}\to\mathbb{R}^{d}$ denotes the flow velocity
vector and $p:\mathbb{R}^{1+d}\to\mathbb{R}$ describes the scalar pressure,
$\Delta_{\perp}:=\partial^{2}_{2}+\cdots+\partial^{2}_{d}$, and
$\varepsilon_{1},\varepsilon_{\perp}\geq 0$ are some constants. In the Ekman
boundary layers for rotating fluids (cf. [12]), it makes sense to consider the
different viscosities of the type $\varepsilon_{1}\geq 0$ and
$\varepsilon_{\perp}>0$. If $\varepsilon_{1}=\varepsilon_{\perp}=0$, then
(8.1) is the Euler equation. As usual, (8.1) may be rewritten in the
equivalent form:

$\displaystyle
u_{t}-\varepsilon_{1}\partial_{1}^{2}u-\varepsilon_{\perp}\Delta_{\perp}u+\mathbb{P}\
\textrm{div}(u\otimes u)=0,$ (8.2)

where $\mathbb{P}:=I-{\rm div}\,\Delta^{-1}\nabla$ is the projection onto the
divergence free vector fields, $I\in M(d)$ is the identity matrix. We need to
extend the equation (8.2) to the complex values
$u:\mathbb{R}^{1+d}\to\mathbb{C}^{d}$, since otherwise our solution space
contains only the trivial $0$. We may also extend the coefficients to
$\varepsilon_{1},\varepsilon_{\perp}\in\mathbb{C}$.

Since the symbols for
$L:=\varepsilon_{1}\partial_{1}^{2}+\varepsilon_{\perp}\Delta_{\perp}$ and
$N:=\mathbb{P}\operatorname{div}$ are given by

$\begin{split}(e^{t\hat{L}}\hat{N})_{jk}=e^{-t\varepsilon_{1}\xi_{1}^{2}-t\varepsilon_{\perp}|\xi^{\perp}|^{2}}i(\xi_{k}-\xi_{j})\quad(j,k=1,\ldots,d),\end{split}$
(8.3)

the condition (C-s) is satisfied with $\mathrm{v}=e_{1}$ and $\theta=1/2$ if
$\varepsilon_{\perp}>0$. Hence, Corollary 4.2 implies that the Cauchy problem
for (8.2) is globally wellposed in $\mathcal{F}^{-1}\mathcal{X}^{d}$ for
$t\geq 0$, for any $d\geq 2$, $\varepsilon_{1}\in\mathbb{C}$ and
$\varepsilon_{\perp}>0$. If both $\varepsilon_{1},\varepsilon_{\perp}>0$, then
it does not depend on the direction of the half space: For any
$\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$, it is globally wellposed in
$\mathcal{F}^{-1}\mathcal{X}(\mathrm{v})^{d}$ on $t\geq 0$.

For more general $\varepsilon_{\perp}\in\mathbb{C}$, let $R\in C^{+}_{0}$ be
super-additive and $\mathcal{R}:=\\{\xi\in\mathbb{R}^{d};|\xi|\leq
R(\mathrm{v}\cdot\xi)\\}$. Then for any $d\geq 2$ and
$\varepsilon_{1},\varepsilon_{\perp}\in\mathbb{C}$, Theorem 1.2 implies that
the Cauchy problem for (8.2) is globally wellposed in
$\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}(\mathrm{v})^{d}$ for
$t\in\mathbb{R}$. The simplest choice of $\mathcal{R}$ is
$\\{|\xi|\leq\lambda\mathrm{v}\cdot\xi\\}$ for some constant $\lambda>0$, but
it may be much bigger as $\mathrm{v}\cdot\xi\to\infty$.

Let $\mathfrak{p}(\xi):=C[(\mathrm{v}\cdot\xi)+(\mathrm{v}\cdot\xi)^{2}]$ for
some constant $C>0$. If $C$ is large enough, depending only on
$\varepsilon_{1},\varepsilon_{\perp},\mathrm{v}$, then (7.2) is satisfied for
some $0<c_{0}\leq 1\leq C_{0}<\infty$. Then Theorem 7.1 implies that for any
$s<0$, $a\geq 2$, $Z\in\mathcal{Z}^{1,s}_{\mathfrak{p}}(\mathcal{R})$, and any
$e^{s\mathfrak{p}^{a}}\hat{u}_{0}\in Z$ with
$\operatorname{supp}\hat{u}_{0}\subset\mathcal{R}$, there exists some $b\geq
1$ such that (8.2) has a unique global solution $u$ satisfying
$e^{b(s\mathfrak{p}^{a}-t\mathfrak{p})}\hat{u}(t)\in[Z(L^{\infty}_{t}(0,\infty))\cap
C([0,\infty);Z)]^{d}$. See (LABEL:L1-Z) and (LABEL:anis-Z) for examples of
$Z$.

For the 3D Navier-Stokes ($d=3$, $\varepsilon_{1}=\varepsilon_{\perp}>0$), Li
and Sinai [22] constructed complex-valued blow-up solutions with Fourier
support on a ball far from the origin and inside a cone. More precisely, they
constructed a set of initial data with 10 parameters for which the energy and
the enstrophy (namely the $L^{2}$ and $\dot{H}^{1}$ norms) of the solution
blow up in finite time. Since their initial data satisfy our conditions, it is
another and highly non-trivial example of the global wellposedness beyond the
classical blow-up. Although the inductive construction of the solution works
in the same way as for the simple example in Section 5.1, the main difficulty
in the Navier-Stokes equation comes from the projection $\mathbb{P}$, which
eliminates the major part of the quadratic interaction under the Fourier
support restriction. Similar blow-up sets of initial data have been
constructed also for the 2D Burgers equation and other hydrodynamical models
[23, 24].

###  8.2. Compressible Navier-Stokes and Euler equations

Next we consider the compressible Navier-Stokes equations for the ideal gas
(cf. Lions [25])

$\left\\{\begin{array}[]{l}\partial_{t}\varrho+\operatorname{div}(\varrho
u)=0,\\\ \partial_{t}(\varrho u)+\operatorname{div}(\varrho u\otimes u)+\nabla
p=\mathcal{L}_{\alpha,\beta}u:=(\alpha+\beta)\nabla\operatorname{div}u+\beta\Delta
u,\\\ \partial_{t}(\varrho e)+\operatorname{div}(\varrho
ue)+p\operatorname{div}u-\kappa\Delta
T=b_{\alpha,\beta}(u):=\tfrac{\beta}{2}\operatorname{tr}(\nabla u+(\nabla
u)^{\top})^{2}+\alpha(\operatorname{div}u)^{2},\end{array}\right.$

where $u:\mathbb{R}^{1+d}\to\mathbb{R}^{d}$ is the velocity,
$\varrho:\mathbb{R}^{1+d}\to[0,\infty)$ is the density,
$T:\mathbb{R}^{1+d}\to[0,\infty)$ is the temperature, $p=R\varrho T$ is the
pressure, $e=c_{v}T$ is the internal energy, for some constants
$\alpha,\beta,\kappa,R,c_{v}>0$. If $\alpha=\beta=\kappa=0$, then they are the
compressible Euler equations. If $\varrho\not=0$, then the equations of
momentum and energy may be rewritten as

$\begin{split}&\partial_{t}u+u\cdot\nabla u+R(\nabla
T+T\tfrac{\nabla\varrho}{\varrho})=\tfrac{1}{\varrho}\mathcal{L}_{\alpha,\beta}u,\\\
&\partial_{t}T+u\cdot\nabla
T+\tfrac{R}{c_{v}}T\operatorname{div}u-\tfrac{\kappa}{c_{v}\varrho}\Delta
T=\tfrac{1}{c_{v}\varrho}b_{\alpha,\beta}(u).\end{split}$ (8.4)

Let $\varrho=\rho_{*}+\rho$ for some constant $\rho_{*}>0$. Then the system is
rewritten for $(\rho,u,T)$ as

$\begin{split}\begin{cases}\partial_{t}\rho+\rho_{*}\operatorname{div}u=-\operatorname{div}(\rho
u),\\\ \partial_{t}u-\tfrac{1}{\rho_{*}}\mathcal{L}_{\alpha,\beta}u+R\nabla
T=-u\cdot\nabla
u-\tfrac{R}{\rho_{*}+\rho}T\nabla\rho-\tfrac{\rho}{\rho_{*}(\rho_{*}+\rho)}\mathcal{L}_{\alpha,\beta}u,\\\
\partial_{t}T-\tfrac{\kappa}{c_{v}\rho_{*}}\Delta T=-u\cdot\nabla
T-\tfrac{R}{c_{v}}T\operatorname{div}u+\tfrac{1}{c_{v}(\rho_{*}+\rho)}b_{\alpha,\beta}(u)-\tfrac{\kappa\rho}{c_{v}\rho_{*}(\rho_{*}+\rho)}\Delta
T.\end{cases}\end{split}$ (8.5)

Because of the quasi-linear terms ($\mathcal{L}_{\alpha,\beta}u$ and $\Delta
T$ on the right side), the condition (C-s) is not satisfied as in the
incompressible case, but we may still apply Theorems 1.2 and 7.1 in the same
way for global wellposedness in
$\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}(\mathrm{v})^{d}$. Note that the
singularity at $\rho=-\rho_{*}$ does not matter in our setting, without any
size restriction (cf. Sections 5.1–5.2). We may also take more general
constants $\alpha,\beta,\kappa,R,1/c_{v}\in\mathbb{C}$.

The same argument works also for the equations of $(\rho,u,\tau)$ with
$T=T_{*}+\tau$ for some constant $T_{*}>0$. We may similarly consider the
equations for the isentropic gas, where $p=a\varrho^{\gamma}$ for some
constants $a,\gamma>0$. In this case, the equations become for $(\rho,u)$

$\begin{split}\begin{cases}\partial_{t}\rho+\rho_{*}\operatorname{div}u=-\operatorname{div}(\rho
u),\\\
\partial_{t}u-\tfrac{1}{\rho_{*}}\mathcal{L}_{\alpha,\beta}u+\tfrac{a\gamma}{\rho_{*}^{2-\gamma}}\nabla\rho=-u\cdot\nabla
u+[\tfrac{a\gamma}{\rho_{*}^{2-\gamma}}-\tfrac{a\gamma}{(\rho_{*}+\rho)^{2-\gamma}}]\nabla\rho-\tfrac{\rho}{\rho_{*}(\rho_{*}+\rho)}\mathcal{L}_{\alpha,\beta}u,\end{cases}\end{split}$
(8.6)

to which Theorems 1.2 and 7.1 are applicable, with any constants
$\alpha,\beta,\gamma,a\in\mathbb{C}$.

###  8.3. Nonlinear wave and Klein-Gordon equations

Consider the nonlinear Klein-Gordon equation in the form

$\displaystyle\left\\{\begin{array}[]{l}u_{tt}+c^{2}u-\Delta
u=\mathscr{H}(\partial_{x}^{\alpha^{1}}u,\ldots,\partial_{x}^{\alpha^{m}}u,\partial_{x}^{\beta^{1}}u_{t},\ldots,\partial_{x}^{\beta^{n}}u_{t}),\\\
u(0,x)=u_{0}(x),\ \ u_{t}(0,x)=u_{1}(x),\end{array}\right.$ (8.9)

for $u(t,x):\mathbb{R}^{1+d}\to\mathbb{C}$, with a constant $c\geq 0$,
$m,n\in\mathbb{N}$,
$\alpha^{1},\ldots,\alpha^{m},\beta^{1},\ldots,\beta^{n}\in\mathbb{N}_{0}^{d}$
and $\mathscr{H}:\mathbb{C}^{m+n}\to\mathbb{C}$ defined and holomorphic in a
neighborhood of $0$, satisfying $\mathscr{H}(0)=0$. It is the nonlinear wave
equation if $c=0$. The equation may be expanded into a first order system for
$U=(U_{0},\ldots,U_{d+1}):=(u,\nabla u,u_{t})$

$\begin{split}\begin{cases}\partial_{t}U_{0}-U_{d+1}=0,\\\
\partial_{t}U_{k}-\partial_{k}U_{d+1}=0\quad(1\leq k\leq d),\\\
\partial_{t}U_{d+1}+c^{2}U_{0}-\sum_{1\leq k\leq
d}\partial_{k}U_{k}=\tilde{\mathscr{H}}(U),\\\ U(0)=(u_{0},\nabla
u_{0},u_{1}),\end{cases}\end{split}$ (8.10)

where $\tilde{\mathscr{H}}(U)$ is an analytic function of derivatives of $U$
derived from
$\mathscr{H}(\partial^{\alpha^{1}}_{x}u,\ldots,\partial^{\beta^{n}}_{x}u_{t})$,
by replacing $u_{t}$ with $U_{d+1}$ and $\partial^{\alpha}_{x}u$ with
$\partial^{\alpha-e_{k}}_{x}U_{k}$ (if $\alpha\geq e_{k}$) or with $U_{0}$ (if
$\alpha=0$). The linear part of $\tilde{\mathscr{H}}$ should be included in
the operator $L$ in applying our results.

For any solution $U$ of the above system, the equations imply
$U_{d+1}=\partial_{t}U_{0}$ and $U_{k}=\nabla U_{0}+\varphi$ for some
$\varphi(x)$ independent of $t$. Hence the initial constraint $U_{k}(0)=\nabla
U_{0}(0)$ implies $U_{k}=\nabla U_{0}$ for all time. Then by definition of
$\tilde{\mathscr{H}}$, $U_{0}$ satisfies the original equation (8.9). The
converse relation is obvious from the definition of the system.

If the equation is semilinear in $\xi^{\perp}$, namely $d=1$ or

$\begin{split}\max_{1\leq j\leq m}\sum_{2\leq k\leq d}\alpha^{j}_{k}\leq
1,\quad\max_{1\leq j\leq n}\sum_{2\leq k\leq d}\beta^{j}_{k}=0,\end{split}$
(8.11)

then $\tilde{\mathscr{H}}$ has no derivative in $\xi^{\perp}$, while
$e^{t\hat{L}}$ is bounded on $\xi\in\mathbb{R}^{d}$, including derivatives,
locally in $t\in\mathbb{R}$. Hence the condition (C-s) is satisfied with
$\mathrm{v}=e_{1}$ and $\theta=0$, and Corollary 4.2 implies the global
wellposedness of the system for $U\in\mathcal{F}^{-1}\mathcal{X}^{d+2}$ on
$t\in\mathbb{R}$. If $|\alpha^{j}|\leq 1$ and $\beta^{j}=0$ for all $j$, then
it is independent of the direction of the half space: the global wellposedness
holds in $\mathcal{F}^{-1}\mathcal{X}(\mathrm{v})^{d+1}$ for any
$\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$.

In the general case of arbitrary derivatives $\\{\alpha^{j},\beta^{j}\\}$, we
may apply Theorems 1.2 and 7.1 to the above system, which imply global
wellposedness in $\mathcal{F}^{-1}\mathcal{X}_{\mathcal{R}}(\mathrm{v})^{d+2}$
and unique global existence in weighted $Z$ with frequencies in the conical
region $\mathcal{R}$ as before.

###  8.4. Nonlinear Schrödinger equations

Similarly to the Klein-Gordon equations, we may consider the nonlinear
Schrödinger equations in the form

$\begin{split}iu_{t}+\Delta
u=\mathscr{H}(\partial_{x}^{\alpha^{1}}u,\ldots,\partial_{x}^{\alpha^{m}}u),\end{split}$
(8.12)

for $u(t,x):\mathbb{R}^{1+d}\to\mathbb{C}$ with the same type of nonlinearity
as above. In this case, the global wellposedness by Corollary 4.2 works if
there is no derivative in the orthogonal directions, namely $\alpha_{k}^{j}=0$
for all $j$ and $k\geq 2$. Otherwise, we may still apply Theorems 1.2 and 7.1
for the global wellposedness on the conical region.

Note that the standard nonlinear Schrödinger equations are not in the above
form, because of the complex conjugate. However, we may consider the equations
in the form

$\begin{split}iu_{t}+\Delta
u=\mathscr{H}(\partial_{x}^{\alpha^{1}}u,\ldots,\partial_{x}^{\alpha^{m}}u,\partial_{x}^{\beta^{1}}u^{\star},\ldots,\partial_{x}^{\beta^{n}}u^{\star}),\end{split}$
(8.13)

where $u^{\star}$ denotes the complex conjugate in the Fourier space, namely

$\begin{split}u^{\star}(t,x):=\overline{u(t,-x)},\quad\mathcal{F}u^{\star}(t,\xi)=\overline{\mathcal{F}u(t,\xi)}.\end{split}$
(8.14)

See [1, 2] for the one-dimensional integrable models, with cubic terms
containing one $u^{\star}$ and at most one derivative. Unique global solutions
were constructed in [7] under the Fourier support restriction in $(0,\infty)$.

The above equation may be expanded to the system of $U=(u,v):=(u,u^{\star})$
with

$\begin{split}\begin{cases}iu_{t}+\Delta
u=\mathscr{H}(\partial_{x}^{\alpha^{1}}u,\ldots,\partial_{x}^{\alpha^{m}}u,\partial_{x}^{\beta^{1}}v,\ldots,\partial_{x}^{\beta^{n}}v),\\\
iv_{t}-\Delta
v=-\bar{\mathscr{H}}((-\partial_{x})^{\alpha^{1}}v,\ldots,(-\partial_{x})^{\alpha^{m}}v,(-\partial_{x})^{\beta_{1}}u,\ldots,(-\partial_{x})^{\beta^{n}}u),\end{cases}\end{split}$
(8.15)

where $\bar{\mathscr{H}}$ is the power series derived from $\mathscr{H}$ by
replacing all the coefficients with their complex conjugate. $v=u^{\star}$ is
an invariant subspace of this system.

We may apply Theorems 1.2 and 7.1 to get global solutions to this system. If
it has no derivative in the orthogonal directions, then Corollary 4.2 is also
applicable for the global wellposedness on the half space. The invariance of
$v=u^{\star}$ is immediate from the uniqueness in those results. In
particular, the simplest model from [1]:

$\begin{split}i\dot{u}+u_{xx}=\alpha u^{2}u^{\star},\end{split}$ (8.16)

for $u(t,x):\mathbb{R}^{2}\to\mathbb{C}$ and a constant $\alpha\in\mathbb{C}$,
is globally wellposed for $t\in\mathbb{R}$ in $\mathcal{F}^{-1}\mathcal{X}$,
as well as in the smaller spaces
$\mathcal{F}^{-1}\mathscr{D}^{\prime}_{\geq\varepsilon}$, for all
$\varepsilon>0$.

If $\alpha\not=0$, this equation has a family of stationary solutions

$\begin{split}u(x)=\tfrac{c}{x+iz}=\mathcal{F}^{-1}[-ic1_{\xi>0}e^{-z\xi}],\quad
c:=(2/\alpha)^{1/2},\quad z\in(0,\infty),\end{split}$ (8.17)

contained in $\mathcal{F}^{-1}\mathcal{X}\cap H^{\infty}(\mathbb{R})$, as well
as in (LABEL:L1-Z) and (LABEL:anis-Z) with $\mathrm{v}=1$. The solutions
$\tfrac{c}{x+iz}$ with $z<0$ are in those spaces with $\mathrm{v}=-1$, while
the spaces with $\mathrm{v}=1$ contain more exotic solutions:
$\tfrac{c}{x+i0}=c(p.v.\tfrac{1}{x}-i\pi\delta)=\mathcal{F}^{-1}[-ic1_{\xi>0}]\in\mathscr{S}^{\prime}(\mathbb{R})$
and
$\mathcal{F}^{-1}[-ic1_{\xi>0}e^{z\xi}]\not\in\mathscr{S}^{\prime}(\mathbb{R})$
with $z>0$.

One may wonder if the same argument would work for the standard equations with
$\bar{u}$ in the nonlinearity. In fact the global results apply to the
extended system of $(u,v):=(u,\bar{u})$, but the intersection of the invariant
subspace $v=\bar{u}$ and $\mathcal{F}^{-1}\mathcal{X}$ is only the trivial $0$
solution.

###  8.5. Completely integrable equations

For nonlinear dispersive equations, it is known that complete integrability
may drastically lower the regularity needed for the Cauchy problem to be
wellposed. In particular, for the KdV equation

$\begin{split}u_{t}+u_{xxx}=6uu_{x}\end{split}$ (8.18)

for $u(t,x):\mathbb{R}^{1+1}\to\mathbb{R}$, Kappeler and Topalov [18] proved
the global wellposedness in $H^{-1}(\mathbb{R}/\mathbb{Z})$, and Killip and
Vişan [20] in $H^{-1}(\mathbb{R})$. On the other hand, Molinet [27, 28] proved
the illposedness in $H^{s}$ for $s<-1$ on both the domains.

For mixed initial data, Tsugawa [33] proved the local wellposedness in a
Banach space of quasi-periodic functions, including initial data in the form

$\begin{split}u_{0}=\sum_{k=1}^{N}f_{k},\quad f_{k}\in
H^{s}(\mathbb{R}/\ell_{k}\mathbb{Z}),\quad s>-\tfrac{1}{2N},\end{split}$
(8.19)

for any fixed $N\in\mathbb{N}$ and $\\{\ell_{k}\\}_{1\leq k\leq
N}\subset(0,\infty)$. For almost periodic data (which forms a closed subspace
in $L^{\infty}(\mathbb{R})$), the unique global solution was constructed by
Binder, Damanik, Goldstein and Lukic [4] under some spectral conditions.
Laurens [21] proved the global wellposedness in $V(t)+H^{-1}(\mathbb{R})$,
around any fixed solution $V\in C(\mathbb{R};H^{5}(\mathbb{R}/\mathbb{Z}))$.

For the complex-valued solutions, Birnir [5] proved blow-up in
$L^{\infty}_{x}$ of the solutions given by Airault, McKean and Moser [3], for
the initial data $u(0,x)=6\wp(x+is)$, where $\wp$ is the Weierstrass elliptic
function with the periods $1,i\omega$ for some $\omega>0$, and
$s\in(-\omega,\omega)$. The Fourier support is not within $[0,\infty)$. The
solution is in the form

$\begin{split}u(t,x)=2[\wp(x+2\mathop{\mathrm{Re}}z(t)+is)+\wp(x-z(t)+is)+\wp(x-\bar{z}(t)+is)]\end{split}$
(8.20)

for some smooth $z:\mathbb{R}\to\mathbb{C}$ satisfying $z(0)=0$, and the blow-
up occurs when $\mathop{\mathrm{Im}}z(t)\in\pm s+\omega\mathbb{Z}$ due to the
poles of $\wp$. Similarly to Remark 5.2, such singularities are not continuous
in $\mathcal{F}^{-1}\mathscr{D}^{\prime}(\mathbb{R})$.

On the other hand, Corollary 4.2 yields the global wellposedness for
$t\in\mathbb{R}$ in $\mathcal{F}^{-1}\mathcal{X}$, where more wild initial
data are allowed, under the Fourier support constraint. Indeed,
$\mathcal{F}^{-1}\mathcal{X}$ includes all initial data in the form

$\begin{split}u_{0}=\sum_{n\in\mathbb{Z}}f_{n},\quad\begin{cases}f_{0}\in
H^{s_{0}}(\mathbb{R}),\
\operatorname{supp}\mathcal{F}f_{0}\subset[0,\infty),&(n=0),\\\ f_{n}\in
H^{s_{n}}(\mathbb{R}/\ell_{n}\mathbb{Z}),\
\operatorname{supp}\mathcal{F}f_{n}\subset(0,\infty)&(n\not=0),\end{cases}\end{split}$
(8.21)

for any sequences $\\{s_{n}\\}_{n\in\mathbb{Z}}\subset\mathbb{R}$ and
$\\{\ell_{n}\\}_{n\in\mathbb{Z}}\subset(0,\infty)$, satisfying

$\begin{split}\lim_{n\to\infty}\ell_{n}=0,\quad\inf_{\delta>0}\sum_{n<0}\|\mathcal{F}f_{n}\|_{\mathcal{M}(\mathbb{R}^{d}_{<\delta})}<\infty.\end{split}$
(8.22)

Theorem 7.1 also yields unique global solutions.

Similarly to the previous equation (8.16), the KdV has stationary solutions

$\begin{split}u(x)=\tfrac{1}{(x+iz)^{2}}=-\mathcal{F}^{-1}[1_{\xi>0}\xi
e^{-z\xi}]\quad(z\in\mathbb{C},\ \mathop{\mathrm{Re}}z>0)\end{split}$ (8.23)

contained in $\mathcal{F}^{-1}\mathcal{X}\cap H^{\infty}(\mathbb{R})$ and
other spaces. The solutions $\tfrac{1}{(x+iz)^{2}}$ with
$\mathop{\mathrm{Re}}z<0$ are in the spaces with $\mathrm{v}=-1$, while the
spaces with $\mathrm{v}=1$ contain also
$\tfrac{1}{(x-a+i0)^{2}}\in\mathscr{S}^{\prime}(\mathbb{R})$ for
$a\in\mathbb{R}$ and $\mathcal{F}^{-1}[1_{\xi>0}\xi
e^{-z\xi}]\not\in\mathscr{S}^{\prime}(\mathbb{R})$ for
$\mathop{\mathrm{Re}}z<0$.

The same story holds for some other integrable equations, such as the modified
KdV equation and the Benjamin-Ono equation. Since our wellposedness results
require little for the equations, they are convenient for equations derived
from algebraic viewpoints. For example, consider the deformed AKNS system on
$\mathbb{R}^{2}$

$\displaystyle\left\\{\begin{array}[]{l}u_{t}-u_{xx}=2u^{2}v+u^{2}v(vu_{yy}+2u_{y}v_{y})+2u(vu_{xy}+v_{x}u_{y}),\\\
v_{t}+v_{xx}=-2v^{2}u-v^{2}u(uv_{yy}+2u_{y}v_{y})-2v(uv_{xy}+u_{x}v_{y}),\\\
u(0,x)=u_{0}(x),\ \ v(0,x)=v_{0}(x),\end{array}\right.$ (8.27)

for $(u,v)(t,x,y):\mathbb{R}^{1+2}\to\mathbb{C}^{2}$, where the second
equation is a backward parabolic equation. (8.27) is Lax integrable and
symmetry integrable, cf. [26]. Theorems 1.2 and 7.1 apply to the above system
as well. Similarly, we may apply those results to the integrable
$(3+1)$-dimensional KdV equation in [17]:

$\begin{split}&u_{t}-u_{xxx}=3bu_{z}(u_{x}+a_{1}uu_{y}+a_{2}u^{2}u_{z})^{2}\\\
&\qquad+\partial_{x}(-2u^{3}+\tfrac{3}{2}a^{2}u^{2}u_{yy}+3auu_{xy}+3bu^{2}u_{xz}+\tfrac{3}{2}b^{2}u^{4}u_{zz}+3abu^{3}u_{zy})\\\
&\qquad+a\partial_{y}(a^{2}u^{3}u_{yy}-u^{4}+\tfrac{3}{2}au^{2}u_{xy}+\tfrac{9}{4}abu^{4}u_{zy}+\tfrac{6}{5}b^{2}u^{5}u_{zz}+2bu^{3}u_{xz})\\\
&\qquad+b\partial_{z}(b^{2}u^{6}u_{zz}-\tfrac{3}{5}au^{5}+\tfrac{3}{2}bu^{4}u_{xz}+\tfrac{3}{4}a^{2}u^{4}u_{yy}+\tfrac{9}{5}abu^{5}u_{zy}+au^{3}u_{xy}),\end{split}$
(8.28)

where $u(t,x,y,z):\mathbb{R}^{1+3}\to\mathbb{C}$ and
$a,b,a_{1},a_{2}\in\mathbb{C}$ are some constants.

###  8.6. Nonlinear hyperbolic conservation laws

The general systems of nonlinear hyperbolic conservation laws on
$\mathbb{R}^{d}$

$\displaystyle\partial_{t}u_{j}+\sum^{d}_{k=1}\partial_{k}\mathscr{H}_{jk}(u)=0\quad(j=1,\ldots,n),\
\ $ (8.29)

for $u:\mathbb{R}^{1+d}\to\mathbb{C}^{n}$, may be also treated by our results,
if each $\mathscr{H}_{jk}:\mathbb{C}^{n}\to\mathbb{C}$ is defined and analytic
in some neighborhood of $0\in\mathbb{C}^{n}$. Decomposing $\mathscr{H}$ into
the linear and super-linear parts, one may easily see that Theorems 1.2 and
7.1 apply to the above system. In fact, it does not matter whether the
nonlinearity is in the divergence form.

###  8.7. Nonlinear parabolic equations

Constantin–Lax–Majda [11, 32] proposed the following one-dimensional model of
the vorticity equation:

$\displaystyle v_{t}=\varepsilon v_{xx}+v\mathcal{H}v,$ (8.30)

where $v:\mathbb{R}^{1+1}\to\mathbb{R}$ and
$\mathcal{H}=\mathcal{F}^{-1}(i\operatorname{sign}\xi)\mathcal{F}$ is the
Hilbert transform. In the same way as for the KdV, Corollary 4.2 yields the
global wellposedness in $\mathcal{F}^{-1}\mathcal{X}$, and Theorem 7.1 yields
unique global solutions, both for $t\in\mathbb{R}$ and
$\varepsilon\in\mathbb{C}$. It has also the stationary solutions in the same
form, $v(x)=-c\mathcal{F}^{-1}[1_{\xi>0}\xi e^{-z\xi}]$ with
$c:=6i\varepsilon$.

More generally, we may consider the fractional heat equation

$\displaystyle
u_{t}=\varepsilon(-\Delta)^{s/2}u+\mathscr{H}(\partial_{x}^{\alpha^{1}}u,\ldots,\partial_{x}^{\alpha^{m}}u),$
(8.31)

for $u:\mathbb{R}^{1+d}\to\mathbb{C}$, where $\varepsilon,s\in\mathbb{C}$,
$m\in\mathbb{N}$, $\alpha^{1},\ldots,\alpha^{m}\in\mathbb{N}_{0}^{d}$ are
constants and $\mathscr{H}:\mathbb{C}^{m}\to\mathbb{C}$ is holomorphic in a
$0$-neighborhood, satisfying $\mathscr{H}(0)=0$. Here the boundedness of
$L=(-\Delta)^{s/2}$ as $\xi_{1}\to 0$ requires $\mathop{\mathrm{Re}}s\geq 0$.
If $\mathop{\mathrm{Re}}\varepsilon>0$ and

$\begin{split}\bar{\alpha}:=\max_{1\leq j\leq m}\sum_{2\leq k\leq
d}\alpha_{k}^{j}<\mathop{\mathrm{Re}}s,\end{split}$ (8.32)

then the condition (C-s) with $\theta=\bar{\alpha}/\mathop{\mathrm{Re}}s$ and
$\mathrm{v}=e_{1}$ is satisfied by the equation of
${\langle\nabla\rangle}^{\bar{\alpha}}u$, so Corollary 4.2 yields the global
wellposedness for
$u\in{\langle\nabla\rangle}^{-\bar{\alpha}}\mathcal{F}^{-1}\mathcal{X}$ on
$t\geq 0$. If $\tilde{\alpha}:=\max_{1\leq j\leq
m}|\alpha^{j}|<\mathop{\mathrm{Re}}s$, then it is independent of the
direction: the global wellposedness holds for
$u\in{\langle\nabla\rangle}^{-\tilde{\alpha}}\mathcal{F}^{-1}\mathcal{X}(\mathrm{v})$
on $t\geq 0$ for any $\mathrm{v}\in\mathbb{R}^{d}\setminus\\{0\\}$. Theorem
1.2 is applicable without any restriction on $\\{\alpha^{j}\\}$ and
$\varepsilon\in\mathbb{C}$ for $\mathop{\mathrm{Re}}s\geq 0$. If
$\mathop{\mathrm{Re}}s\geq 1$, then the decay condition of $\hat{L}$ in (7.2)
is satisfied, so that Theorem 7.1 can also be applied.

We may also consider nonlinear diffusion, such as the porous medium equation:

$\begin{split}&u_{t}=\Delta(u^{m}),\end{split}$ (8.33)

or the fast diffusion equation, and the $p$-Laplace heat equation:

$\begin{split}u_{t}=\operatorname{div}(|\nabla u|^{p-2}\nabla u),\quad|\nabla
u|^{p-2}:=[\sum_{j=1}^{d}(\partial_{j}u)^{2}]^{p/2-1}.\end{split}$ (8.34)

Here $u(t,x):\mathbb{R}^{1+d}\to\mathbb{R}$ and $m,p>0$ are given constants.

In fact, if $m\geq 2$ is an even integer, or $p\geq 4$ is an even integer,
then the right side is already in the nonlinear form of our setting, after
extending $u:\mathbb{R}^{1+d}\to\mathbb{C}$, so that we may apply Theorems 1.2
and 7.1.

In the remaining case of $m$ or $p$, we may expand the power around a fixed
constant. For the porous medium, let $u=u_{*}+v$ for some constant
$u_{*}\in\mathbb{C}\setminus\\{0\\}$. Then the equation is rewritten for $v$
as

$\begin{split}v_{t}-mu_{*}^{m-1}\Delta
v=\Delta\mathscr{H}_{m}^{u_{*}}(v),\quad\mathscr{H}_{m}^{u_{*}}(v):=\sum_{n\geq
2}\tfrac{\Gamma(m+1)u_{*}^{m-n}}{\Gamma(m-n+1)n!}v^{n},\end{split}$ (8.35)

which is in the form to which Theorems 1.2 and 7.1 apply. For the
$p$-Laplacian, let $u=u_{*}+\xi_{*}\cdot x+v$ for some constants
$u_{*}\in\mathbb{C}$ and $\xi_{*}\in\mathbb{C}^{d}\setminus\\{0\\}$. Then the
equation is rewritten for $v$

$\begin{split}&v_{t}-|\xi_{*}|^{p-2}\Delta
v-(p-2)|\xi_{*}|^{p-4}\operatorname{div}(\xi_{*}\cdot\nabla
v)=\operatorname{div}\mathscr{H}_{p,\xi_{*}}(\nabla v),\\\
&\quad\mathscr{H}_{p,\xi_{*}}(V):=\mathscr{H}_{p/2-1}^{|\xi_{*}|^{2}}(2\xi_{*}\cdot
V+|V|^{2})(\xi_{*}+V)+(p-2)|\xi_{*}|^{p-4}(\xi_{*}\cdot V)V,\end{split}$
(8.36)

to which we may apply Theorems 1.2 and 7.1. Of course, we may consider various
combinations of those nonlinear terms.

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