# Construction and Stability of type I blowup solutions for non-variational
semilinear parabolic systems

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider in this note the semilinear heat system

$\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,$

where the nonlinearity has no gradient structure taking of the particular form

$f(v)=v|v|^{p-1}\quad\textup{and}\quad g(u)=u|u|^{q-1}\quad\textup{with}\quad
p,q>1,$

or

$f(v)=e^{pv}\quad\textup{and}\quad g(u)=e^{qu}\quad\textup{with}\quad p,q>0.$

We exhibit type I blowup solutions for this system and give a precise
description of its blowup profiles. The method relies on two-step procedure:
the reduction of the problem to a finite dimensional one via a spectral
analysis, then solving the finite dimensional problem by a classical
topological argument based on index theory. As a consequence of our technique,
the constructed solutions are stable under a small perturbation of initial
data. The results and the main arguments presented in this note can be found
in our papers [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag, Ghoul et
al.(2018d)Ghoul, Nguyen, and Zaag].

###### Key words and phrases:

Blowup solution, Blowup profile, Stability, Semilinear parabolic system

###### 1991 Mathematics Subject Classification:

Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

H. Zaag is supported by the ANR project ANAÉ ref. ANR-13-BS01-0010-03.
——————–

Tej-Eddine Ghoul†, Van Tien Nguyen† and Hatem Zaag∗

†New York University in Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab
Emirates.

∗Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93430,
Villetaneuse, France.

##  1\. Introduction.

In [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag, Ghoul et al.(2018d)Ghoul,
Nguyen, and Zaag], we consider the semilinear parabolic system

$\left\\{\begin{array}[]{rl}\partial_{t}u&=\;\;\Delta u+f(v),\\\
\partial_{t}v&=\mu\Delta
v+g(u),\end{array}\right.\quad(x,t)\in\mathbb{R}^{+}\times\mathbb{R}^{N},$
(1.1)

with $N\geq 1$, $\mu>0$ and
$\big{(}u(0),v(0)\big{)}=\big{(}u_{0},v_{0}\big{)}$, where
$(u,v)(t):x\in\mathbb{R}^{N}\to\mathbb{R}^{2}$ and the nonlinearity has no
gradient structure taking of the particular form

$f(v)=v|v|^{p-1}\quad\textup{and}\quad g(u)=u|u|^{q-1}\quad\textup{with}\quad
p,q>1,$ (1.2)

or

$f(v)=e^{pv}\quad\textup{and}\quad g(u)=e^{qu}\quad\textup{with}\quad p,q>0.$
(1.3)

System (1.1) represents a simple model of a reaction-diffusion system
describing heat propagation in a two-component combustible mixture and, as
such, it has been the subject of intensive investigation from the last two
decades (see [Souplet(2005)], [Zheng et al.(2002)Zheng, Zhao, and Chen] and
references therein). We are here mainly interested in proving the existence
and stability of finite time blowup solutions satisfying some prescribed
asymptotic behavior. By finite time blowup, we mean that $T=T(u_{0},v_{0})$,
the maximal existence time of the classical solution $(u,v)$ of problem (1.1),
is finite, and the solution blows up in finite time $T$ in the sense that

$\lim_{t\to
T}\big{(}\|u(t)\|_{L^{\infty}(\mathbb{R}^{N})}+\|v(t)\|_{L^{\infty}(\mathbb{R}^{N})}\big{)}=+\infty.$

Moreover, a finite blowup solution $(u,v)$ of system (1.1) is called Type I if
there exists some positive constant $C$ such that

$\|u(t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq
C\bar{u}(t),\quad\|v(t)\|_{L^{\infty}(\mathbb{R}^{N})}\leq C\bar{v}(t),$ (1.4)

where $(\bar{u},\bar{v})(t)$ is the unique positive blowup solution of the
ordinary differential system associated to (1.1), namely that

$\displaystyle\bar{u}(t)=\Gamma(T-t)^{-\alpha},\quad\bar{v}(t)=\gamma(T-t)^{-\beta}$
$\displaystyle\quad\textup{for \eqref{def:fg1}},$
$\displaystyle\bar{u}(t)=\ln\left[\big{(}p(T-t)\big{)}^{\frac{-1}{q}}\right],\quad\bar{v}(t)=\ln\left[\big{(}q(T-t)\big{)}^{\frac{-1}{p}}\right]$
$\displaystyle\quad\textup{for \eqref{def:fg2}},$

where $(\Gamma,\gamma)$ is determined by

$\gamma^{p}=\alpha\Gamma,\quad\Gamma^{q}=\gamma\beta,\quad\alpha=\frac{p+1}{pq-1},\quad\beta=\frac{q+1}{pq-1}.$
(1.5)

Otherwise, the blowup solution is of Type II.

As for system (1.1)-(1.2) with $\mu=1$, the existence of finite time blowup
solutions was derived by Friedman-Giga [Friedman and Giga(1987)], Escobedo-
Herrero [Escobedo and Herrero(1991a)] (see also [Escobedo and Herrero(1991b)],
[Escobedo and Herrero(1993)], etc). From Andreucci-Herrero-Velázquez
[Andreucci et al.(1997)Andreucci, Herrero, and Velázquez], we know that
estimate (1.4) holds true if

$pq>1,\quad q(pN-2)_{+}<N+2\quad\textup{or}\quad p(qN-2)_{+}<N+2.$

See also Caristi-Mitidieri [Caristi and Mitidieri(1994)], Deng [Deng(1996)],
Fila-Souple [Fila and Souplet(2001)] for more results relative to estimate
(1.4). Knowing that the solution exhibits Type I blowup, the authors of
[Andreucci et al.(1997)Andreucci, Herrero, and Velázquez] were able to obtain
more information about the asymptotic behavior of the solution near the
singularity. Their results were later improved by Zaag [Zaag(2001)]. When
$\mu\neq 1$, much less result has been known, a part from Mahmoudi-Souplet-
Tayachi [Mahmoudi et al.(2015)Mahmoudi, Souplet, and Tayachi] who establishes
a single point blowup result that improves the one obtained in [Friedman and
Giga(1987)]. As for system (1.1) coupled with the nonlinearity (1.3), the only
known result is due to Souplet-Tayachi [Souplet and Tayachi(2016)] who adapted
the technique developed in [Mahmoudi et al.(2015)Mahmoudi, Souplet, and
Tayachi] to obtain the single point blowup result for a class of radially
descreasing solutions. To our knowledge, there are no results concerning the
asymptotic behavior, even for the equidiffusive case, i.e. $\mu=1$. Also we
recall that the study of the non-equidiffusive parabolic system (1.1) ($\mu$
may or may not equal to $1$) are in particular much more involved, both in
terms of behavior of solutions and at the technical level.

In this note we exhibit Type I blowup solutions for system (1.1) and give the
first complete description of its asymptotic behavior. More precisely, we
prove in [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag] the following theorem.

######  Theorem 1.1 (Type I blowup solutions for (1.1)-(1.2) and its
asymptotic behavior, [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag]).

Let $a\in\mathbb{R}^{N}$ and $T>0$. There exist initial data $(u_{0},v_{0})\in
L^{\infty}(\mathbb{R}^{N})\times L^{\infty}(\mathbb{R}^{N})$ for which system
(1.1)-(1.2) has the unique solution $(u,v)$ defined on
$\mathbb{R}^{N}\times[0,T)$ such that

  * (i)

The solution $(u,v)$ blows up in finite time $T$ at the only point $a$.

  * (ii)

(Asymptotic profile) There holds for all $t\in[0,T)$,

$\left\|(T-t)^{\alpha}u(x,t)-\Phi^{*}(z)\right\|_{L^{\infty}(\mathbb{R}^{N})}+\left\|(T-t)^{\beta}v(x,t)-\Psi^{*}(z)\right\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{C}{\sqrt{(T-t)}},$
(1.6)

where $z=\frac{x-a}{\sqrt{(T-t)|\log(T-t)|}}$ and the profiles $\Phi_{0}$ and
$\Psi_{0}$ are explicitly given by

$\forall
z\in\mathbb{R}^{N},\quad\Phi^{*}(z)=\Gamma\big{(}1+b|z|^{2}\big{)}^{-\alpha},\quad\Psi^{*}(z)=\gamma\big{(}1+b|z|^{2}\big{)}^{-\beta},$
(1.7)

with $\Gamma,\gamma,\alpha,\beta$ being introduced in (1.5) and

$b=b(\mu,p,q)=\frac{(pq-1)(2pq+p+q)}{4pq(p+1)(q+1)(\mu+1)}>0.$ (1.8)

  * (iii)

(Final blowup profile) For all $x\neq a$,
$\big{(}u(x,t),v(x,t)\big{)}\to\big{(}u^{*}(x),v^{*}(x)\big{)}\in\Big{[}\mathcal{C}^{2}\big{(}\mathbb{R}^{N}\setminus\\{0\\}\big{)}\Big{]}^{2}$
with

$u^{*}(x)\sim\Gamma\left(\frac{b|x-a|^{2}}{2\big{|}\log|x-a|\big{|}}\right)^{-\frac{p+1}{pq-1}},\quad
v^{*}(x)\sim\gamma\left(\frac{b|x-a|^{2}}{2\big{|}\log|x-a|\big{|}}\right)^{-\frac{q+1}{pq-1}}$
(1.9)

as $|x-a|\to 0$.

######  Remark 1.2.

The asymptotic profile defined in (1.7) with $\mu=1$ is the one among the
classification result established in [Andreucci et al.(1997)Andreucci,
Herrero, and Velázquez] (see also [Zaag(2001)]). This is to say that we can
construct Type I blowup solutions for (1.1)-(1.2) verifying the other
asymptotic profiles described as in [Andreucci et al.(1997)Andreucci, Herrero,
and Velázquez]. However, those constructions would be simpler than our
considered case (1.6) which involves some logarithmic correction to the blowup
variable.

######  Remark 1.3.

The estimate (1.9) is sharp in comparison with the result established in
[Mahmoudi et al.(2015)Mahmoudi, Souplet, and Tayachi] (see Theorem 1.3) where
the authors could only obtain lower pointwise estimates without the
logarithmic correction.

As for system (1.1) coupled with the nonlinearity (1.3), we study in the
special affine space $\mathcal{H}_{\alpha}$ for some positive constant
$\alpha$,

$\mathcal{H}_{\alpha}=\big{\\{}(u,v)\in(\bar{\phi},\bar{\psi})+L^{\infty}(\mathbb{R}^{N})\times
L^{\infty}(\mathbb{R}^{N})\;\textup{where}\;q\bar{\phi}(x)=p\bar{\psi}(x)=-\ln(1+\alpha|x|^{2})\big{\\}},$

and establish in [Ghoul et al.(2018d)Ghoul, Nguyen, and Zaag] the following
result:

######  Theorem 1.4 (Type I blowup solutions for (1.1)-(1.3) and its
asymptotic behavior, [Ghoul et al.(2018d)Ghoul, Nguyen, and Zaag]).

Let $a\in\mathbb{R}^{N}$ and $T>0$. There exist initial data
$(u_{0},v_{0})\in\mathcal{H}_{\alpha}$ for which system (1.1)-(1.3) has the
unique solution $(u,v)$ defined on $\mathbb{R}^{N}\times[0,T)$ such that

  * (i)

The function $(e^{qu},e^{pv})$ blows up in finite time $T$ at the only point
$a$.

  * (ii)

(Asymptotic profile) There holds for all $t\in[0,T)$,

$\left\|(T-t)e^{qu(x,t)}-\Phi_{*}(z)\right\|_{L^{\infty}(\mathbb{R}^{N})}+\left\|(T-t)e^{pv(x,t)}-\Psi_{*}(z)\right\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{C}{\sqrt{(T-t)}},$
(1.10)

where $z=\frac{x-a}{\sqrt{(T-t)|\log(T-t)|}}$ and the profiles $\Phi_{*}$ and
$\Psi_{*}$ are explicitly given by

$\forall z\in\mathbb{R}^{N},\quad
p\Phi_{*}(z)=q\Psi_{*}(z)=\frac{1}{1+b|z|^{2}}\quad\text{with}\;\;b=\frac{1}{2(\mu+1)}.$
(1.11)

  * (iii)

(Final blowup profile) For all $x\neq a$,
$\big{(}u(x,t),v(x,t)\big{)}\to\big{(}u_{*}(x),v_{*}(x)\big{)}\in\Big{[}\mathcal{C}^{2}\big{(}\mathbb{R}^{N}\setminus\\{0\\}\big{)}\Big{]}^{2}$
with

$u_{*}(x)\sim\frac{1}{q}\ln\left(\frac{2b}{p}\frac{\big{|}\log|x-a|\big{|}}{|x-a|^{2}}\right),\quad
v_{*}(x)\sim\frac{1}{p}\ln\left(\frac{2b}{q}\frac{\big{|}\log|x-a|\big{|}}{|x-a|^{2}}\right)\quad\text{as}\;\;|x-a|\to
0.$ (1.12)

######  Remark 1.5.

We can construct Type I blowup solutions for (1.1)-(1.3) satisfying different
blowup profiles that do not have logarithmic correction to the blowup variable
described as in (1.10). This is to say that we can obtain an analogous
classification result for Type I blowup solutions of (1.1)-(1.3) by adapting
the technique of [Andreucci et al.(1997)Andreucci, Herrero, and Velázquez]
with some more technical difficulties.

The proof of Theorem 1.1 and Theorem 1.4 relies on two-step procedure:

  * •

Reduction of an infinite dimensional problem to a finite dimensional one,
through either the spectral analysis of the linearized operator around the
expected profile or the energy-type estimate via the derivation of suitable
Lyapunov functional. Note that the energy-type method breaks down for our
problem because of the non gradient structure of the nonlinearity.

  * •

The control of the finite dimensional problem thanks to a classical
topological argument based on index theory.

This two-step procedure has been successfully applied for various nonlinear
evolution equations to construct both Type I and Type II blowup solutions. It
was the case of the semilinear heat equation treated in [Bricmont and
Kupiainen(1994)], [Merle and Zaag(1997)], [Nguyen and Zaag(2017)] (see also
[Nguyen and Zaag(2016)], [Duong et al.(2018)Duong, Nguyen, and Zaag] for the
case of logarithmic perturbations, [Bressan(1990)], [Bressan(1992)] and [Ghoul
et al.(2017)Ghoul, Nguyen, and Zaag] for the exponential source, [Nouaili and
Zaag(2015)] for the complex-valued case), the Ginzburg-Landau equation in
[Masmoudi and Zaag(2008)], [Nouaili and Zaag(2018)] (see also [Zaag(1998)] for
an earlier work). It was also the nonlinear Schrödinger equation both in the
mass critical [Merle and Raphael(2003), Merle and Raphael(2004), Merle and
Raphael(2005a), Merle and Raphael(2005b)] and mass supercritical [Merle et
al.(2015)Merle, Raphaël, and Rodnianski] cases; the energy critical
[Duyckaerts et al.(2013)Duyckaerts, Kenig, and Merle], [Hillairet and
Raphaël(2012)] and supercritical [C.(2016b)] wave equation; the mass critical
gKdV equation [Martel et al.(2014)Martel, Merle, and Raphaël, Martel et
al.(2015a)Martel, Merle, and Raphaël, Martel et al.(2015b)Martel, Merle, and
Raphaël]; the two dimensional Keller-Segel model [Raphaël and
Schweyer(2014b)]; the energy critical and supercritical geometric equations:
the wave maps [Raphaël and Rodnianski(2012)] and [Ghoul et al.(2018b)Ghoul,
Ibrahim, and Nguyen], the Schrödinger maps [Merle et al.(2013)Merle, Raphaël,
and Rodnianski] and the harmonic heat flow [Raphaël and Schweyer(2013),
Raphaël and Schweyer(2014a)] and [Ghoul et al.(2018a)Ghoul, Ibrahim, and
Nguyen]; the semilinear heat equation in the energy critical [Schweyer(2012)]
and supercritical [C.(2016a)] cases.

As a consequence of our technique, we obtained the following stability result.

######  Theorem 1.6.

The constructed solutions described in Theorem 1.1 and Theorem 1.4 are stable
with respect to initial data.

######  Remark 1.7.

The idea behind the stability result can be formally understood from the
space-time and scaling invariance of the problem as follows: The linearized
operator around the expected profile has two positive eigenvalues
$\lambda_{0}=1,\lambda_{1}=\frac{1}{2}$, a zero eigenvalue $\lambda_{2}=0$,
then a an infinity discrete negative spectrum. From the analysis of stability
of blowup problems, the component corresponding to $\lambda_{0}=1$ has the
exponential growth $e^{s}$, which can be eliminated by a changing of the
blowup time; similarly for the mode $\lambda_{1}=\frac{1}{2}$ by a shifting of
the blowup point; the neutral mode $\lambda_{2}=0$ usually has a polynomial
growth and can be eliminated by using the scaling invariance of the problem.
Since the remaining modes of the linearized operator corresponding to the
negative spectrum decay exponentially, one derive the stability of the
constructed solutions described in Theorem 1.1 and Theorem 1.4. From the
stability result, we expect that the blowup profiles (1.7) and (1.11) are
generic, i.e. the other blowup profiles are unstable. In our opinion, this is
a difficult open question whose a partly particular answer was given by
Herrero-Velázquez [Herrero and Velázquez(1992)] for the one dimensional
semilinear heat equation.

##  2\. A formal computation of the blowup profile.

We brieftly recall in this section the formal approaches in [Ghoul et
al.(2018c)Ghoul, Nguyen, and Zaag, Ghoul et al.(2018d)Ghoul, Nguyen, and Zaag]
to construct a suitable approximate blowup profile for system (1.1). Similar
approaches can be found in [Tayachi and Zaag(2015), Tayachi and Zaag(2016)],
[Ghoul et al.(2017)Ghoul, Nguyen, and Zaag], [Nouaili and Zaag(2018)] and
references therein. The method is based on matched asymptotic expansions which
mainly replies on the spectral properties of the linearized operator around an
expected profile.

#### Similarity variables:

We perform the well known change of variables

$\displaystyle\Phi(y,s)=(T-t)^{\alpha}u(x,t),\quad\Psi(y,s)=(T-t)^{\beta}v(x,t)$
$\displaystyle\quad\textup{for}\;\eqref{def:fg1},$ (2.1)
$\displaystyle\Phi(y,s)=(T-t)e^{qu(x,t)},\quad\Psi(y,s)=(T-t)e^{pv(x,t)}$
$\displaystyle\quad\textup{for}\;\eqref{def:fg2},$ (2.2)

where $\alpha,\beta$ are introduced in (1.5) and

$y=\frac{x}{\sqrt{T-t}},\quad s=-\log(T-t).$

In this way, $(\Phi,\Psi)$ satisfies the new system

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{s}\Phi&=\mathcal{L}_{1}\Phi-\alpha\Phi+|\Psi|^{p-1}\Psi,\\\
\partial_{s}\Psi&=\mathcal{L}_{\mu}\Psi-\beta\Psi+|\Phi|^{p-1}\Phi,\end{array}\right.$
for (1.2), (2.5)
$\displaystyle\left\\{\begin{array}[]{l}\partial_{s}\Phi=\mathcal{L}_{1}\Phi-\Phi-\dfrac{|\nabla\Phi|^{2}}{\Phi}+q\Phi\Psi,\\\
\partial_{s}\Psi=\mathcal{L}_{\mu}\Psi-\Psi-\mu\dfrac{|\nabla\Psi|^{2}}{\Psi}+p\Phi\Psi,\end{array}\right.$
for (1.3), (2.8)

where

$\mathcal{L}_{\eta}f=\eta\Delta f-\frac{y}{2}\cdot\nabla
f=\frac{\eta}{\rho_{\eta}}\nabla\cdot\big{(}\rho_{\eta}\nabla
f\big{)}\quad\textup{with}\quad\eta\in\\{1,\mu\\},$ (2.9)

is the self-adjoint operator with respect to the Hilbert space
$L^{2}_{\rho_{\eta}}(\mathbb{R}^{N},\mathbb{R})$ equipped with the inner
product

$\big{<}f,g\big{>}_{L^{2}_{\rho_{\eta}}}=\int_{\mathbb{R}^{N}}f(y)g(y)\rho_{\eta}(y)dy\quad\textup{with}\quad\rho_{\eta}(y)=\frac{1}{(4\pi)^{N/2}}e^{-\frac{|y|^{2}}{4\eta}}.$

#### Linearized problem and spectral properties of the associated linearized
operator:

Note that the nonzero constant solutions to systems (2.5) and (2.8) are
$(\Gamma,\gamma)$ and $(1/p,1/q)$ respectively. This suggests the
linearization

$\big{(}\bar{\Phi},\bar{\Psi}\big{)}=\big{(}\Phi-\Gamma,\Psi-\gamma\big{)}\;\;\textup{for
\eqref{def:fg1}}\quad\textup{and}\quad\big{(}\bar{\Phi},\bar{\Psi}\big{)}=\big{(}\Phi-1/p,\Psi-1/q\big{)}\;\;\textup{for
\eqref{def:fg2}},$ (2.10)

where $(\bar{\Phi},\bar{\Psi})$ solves the system

$i=1,2,\quad\partial_{s}\binom{\bar{\Phi}}{\bar{\Psi}}=\left(\mathcal{H}+Mc_{i}\right)\binom{\bar{\Phi}}{\bar{\Psi}}+\binom{Q_{i,1}}{Q_{i,2}},$
(2.11)

where $i=1$ stands for the polynomial nonlinearity (1.2) and $i=2$ for the
exponential case (1.3), $Q_{i,1}$ and $Q_{i,2}$ are built to be quadratic, and
the linear operator $\mathcal{H}$ and matrices $Mc_{i}$’s are defined by

$\mathcal{H}=\left(\begin{matrix}\mathcal{L}_{1}&0\\\
0&\mathcal{L}_{\mu}\end{matrix}\right),\quad
Mc_{1}=\begin{pmatrix}-\alpha&\;p\gamma^{p-1}\\\
q\Gamma^{q-1}&\;-\beta\end{pmatrix},\quad Mc_{2}=\left(\begin{matrix}0&q/p\\\
p/q&0\end{matrix}\right).$ (2.12)

The following lemma gives the spectral properties of $\mathcal{H}+Mc_{i}$.

######  Lemma 2.1 (Diagonalization of $\mathcal{H}+Mc_{i}$).

For all $n\in\mathbb{N}$, there exist polynomials $f_{n},g_{n},\tilde{f}_{n}$
and $\tilde{g}_{n}$ of degree $n$ such that

$\Big{(}\mathcal{H}+Mc_{i}\Big{)}\binom{f_{n}}{g_{n}}=\left(1-\frac{n}{2}\right)\binom{f_{n}}{g_{n}},\quad\Big{(}\mathcal{H}+Mc_{i}\Big{)}\binom{\tilde{f}_{n}}{\tilde{g}_{n}}=\lambda^{-}_{i,n}\binom{\tilde{f}_{n}}{\tilde{g}_{n}},$
(2.13)

where

$\lambda^{-}_{1,n}=-\left(\frac{n}{2}+\frac{(p+1)(q+1)}{pq-1}\right),\quad\lambda^{-}_{2,n}=-\left(1+\frac{n}{2}\right).$

###### Proof.

See Lemma 3.2 in [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag] for the
polynomial case (1.2) and Lemma 2.2 in [Ghoul et al.(2018d)Ghoul, Nguyen, and
Zaag] for the exponential case (1.3). The reader is kindly invited to have a
look at precise formulas of the eigenfunctions as well as a proper definition
of the projection according to these eigenmodes in those papers. ∎

#### Inner expansion:

From Lemma 2.1, we know that $\binom{f_{n}}{g_{n}}_{n\geq 3}$ and
$\binom{\tilde{f}_{n}}{\tilde{g}_{n}}_{n\geq 0}$ correspond to negative
eigenvalues of $\mathcal{H}+Mc_{i}$, therefore, we may consider the following
formal expansion under the radially symmetric assumption of the solution,

$\binom{\bar{\Phi}}{\bar{\Psi}}(y,s)=a_{0}(s)\binom{f_{0}}{g_{0}}(y)+a_{2}(s)\binom{f_{2}}{g_{2}},$
(2.14)

where $|a_{0}(s)|+|a_{2}(s)|\to 0$ as $s\to+\infty$. Plugging this ansatz into
(2.11) and projecting onto $\binom{f_{k}}{g_{k}},k=0,2$ yields the ordinary
differential system

$\left\\{\begin{array}[]{ll}a_{0}^{\prime}&=a_{0}+\mathcal{O}(|a_{0}|^{2}+|a_{2}|^{2}),\\\
a_{2}^{\prime}&=c_{*}a_{2}^{2}+\mathcal{O}(|a_{2}|^{3}+|a_{0}a_{2}|+|a_{0}|^{3}),\end{array}\right.$
(2.15)

where

$c_{*}=\frac{2pq+p+q}{4pq(p+1)(q+1)(\mu+1)}\;\;\textup{for
\eqref{def:fg1}}\quad\textup{and}\quad c_{*}=2pq(\mu+1)\;\;\textup{for
\eqref{def:fg2}}.$

Assume that $|a_{0}(s)|=o(|a_{2}(s)|)$ as $s\to+\infty$, we get

$a_{2}(s)=-\frac{1}{c_{*}s}+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)\quad\textup{and}\quad|a_{0}(s)|=\mathcal{O}\left(\frac{1}{s^{2}}\right)\quad\textup{as}\;\;s\to+\infty.$

From (2.14), (2.10) and the definition of the eigenfuntion
$\binom{f_{2}}{g_{2}}$, we end up with the asymptotic behavior

$\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,s)&=\Gamma\left[1-\frac{p+1}{c_{*}}\frac{|y|^{2}}{s}-\frac{2p(1-\mu)}{c_{*}s}\right]+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right),\\\
\Psi(y,s)&=\gamma\left[1-\frac{q+1}{c_{*}}\frac{|y|^{2}}{s}-\frac{2q(\mu-1)}{c_{*}s}\right]+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right),\end{array}\right.$ for (2.5), (2.18)
$\displaystyle\left\\{\begin{array}[]{l}\Phi(y,s)=\frac{1}{p}\left[1-\frac{pq}{c_{*}}\frac{|y|^{2}}{s}+\frac{2\mu
pq}{c_{*}s}\right]+\mathcal{O}\left(\frac{\log s}{s^{2}}\right),\\\
\Psi(y,s)=\frac{1}{q}\left[1-\frac{pq}{c_{*}}\frac{|y^{2}|}{s}+\frac{2pq}{c_{*}s}\right]+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right),\end{array}\right.$ for (2.8), (2.21)

where the convergence takes place in $L_{\rho_{1}}^{2}(\mathbb{R}^{N})\times
L^{2}_{\rho_{\mu}}(\mathbb{R}^{N})$ as well as uniformly on compact sets by
standard parabolic regularity.

#### Outer expansion:

These above asymptotic expansions provide a relevant blowup variable

$z=\frac{y}{\sqrt{s}}=\frac{x}{\sqrt{(T-t)|\log(T-t)|}}.$

We then try to search an approximate solution to (2.5) (respectively (2.8)) of
the form

$\binom{\Phi}{\Psi}(y,s)=\binom{\Phi_{0}}{\Psi_{0}}(z)+\frac{1}{s}\binom{\Phi_{1}}{\Phi_{1}}(z)+\cdots,$
(2.22)

Plugging this anzats to (2.5) (respectively (2.8)) yields the leading order
system

$\displaystyle-\frac{z}{2}\Phi_{0}^{\prime}-\alpha\Phi_{0}+\Phi_{0}^{p}=0,\quad-\frac{z}{2}\Psi_{0}^{\prime}-\beta\Psi_{0}+\Psi_{0}^{q}=0,$
$\displaystyle\quad\textup{for \eqref{sys:PhiPsi1}},$ (2.23)
$\displaystyle-\frac{z}{2}\Phi_{0}^{\prime}-\Phi_{0}+q\Phi_{0}\Psi_{0}=0,\quad-\frac{z}{2}\Psi_{0}^{\prime}-\Psi_{0}+p\Phi_{0}\Psi_{0}=0,$
$\displaystyle\quad\textup{for \eqref{sys:PhiPsi2}},$ (2.24)

subject to the initial condition

$\big{(}\Phi_{0},\Psi_{0}\big{)}(0)=\big{(}\Gamma,\gamma\big{)}\;\;\textup{for
\eqref{asy:PhiPsi1}}\quad\textup{and}\quad\big{(}\Phi_{0},\Psi_{0}\big{)}(0)=\big{(}1/p,1/q\big{)}\;\;\textup{for
\eqref{asy:PhiPsi2}}.$

The solutions of these system are explicitly given by

$\displaystyle\Phi_{0}(z)=\frac{\Gamma}{(1+b|z|^{2})^{\alpha}},\quad\Psi_{0}(z)=\frac{\gamma}{(1+b|z|^{2})^{\beta}}$
$\displaystyle\quad\textup{for \eqref{sys:ode1}},$ (2.25)
$\displaystyle\Phi_{0}(z)=\frac{1}{p(1+b|z|^{2})},\quad\Psi_{0}(z)=\frac{1}{q(1+b|z|^{2})}$
$\displaystyle\quad\textup{for \eqref{sys:ode2}},$ (2.26)

where $b>0$ is an integration constant. By matching the asymptotic expansions
(2.25) with (2.18) and (2.26) with (2.21), we obtain precisely the value of
the constant $b$ as stated in Theorems 1.1 and 1.4 respectively.

In conclusion, we have formally derived the following approximate blowup
profile:

$\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,s)&\sim\varphi(y,s)=\Phi_{0}\left(\frac{y}{\sqrt{s}}\right)-\frac{2\Gamma
p(1-\mu)}{c_{*}s},\\\
\Psi(y,s)&\sim\psi(y,s)=\Psi_{0}\left(\frac{y}{\sqrt{s}}\right)-\frac{2\gamma
q(\mu-1)}{c_{*}s},\end{array}\right.$ for (2.5), (2.29)
$\displaystyle\left\\{\begin{array}[]{ll}\Phi(y,s)=&\sim\varphi(y,s)=\Phi_{0}\left(\frac{y}{\sqrt{s}}\right)+\frac{2\mu
q}{c_{*}s},\\\
\Psi(y,s)=&\sim\psi(y,s)=\Psi_{0}\left(\frac{y}{\sqrt{s}}\right)+\frac{2p}{c_{*}s},\end{array}\right.$
for (2.8). (2.32)

##  3\. The existence proof without technical details.

We present all main arguments of the existence proof without technical details
for which we kindly refer the interested reader to our papers [Ghoul et
al.(2018c)Ghoul, Nguyen, and Zaag, Ghoul et al.(2018d)Ghoul, Nguyen, and
Zaag]. We first deal with the polynomial case (1.2), i.e. the proof of Theorem
1.1, then the exponential case (1.3), i.e. the proof of Theorem 1.4, which is
more delicate due to the presence of the terms $\frac{|\nabla\Phi|^{2}}{\Phi}$
and $\frac{|\nabla\Psi|^{2}}{\Psi}$ in the similarity variables setting (see
(2.8)).

###  3.1. The polynomial case (1.2).

This subsection is devoted to the proof of part $(ii)$ of Theorem 1.1. Parts
$(i)$ and $(iii)$ are consequences of part $(ii)$. The reader can find all
details of the proof in [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag].

#### Formulation of the problem:

In view of the similarity variables (2.1), we see that constructing blowup
solutions for (1.1) coupled with (1.2) satisfying the asymptotic behavior
(1.6) is equivalent to constructing for (2.5) a global in time solution
$(\Phi,\Psi)$ such that

$\sup_{y\in\mathbb{R}^{N}}\Big{(}\big{|}\Phi(y,s)-\Phi^{*}(y/\sqrt{s})\big{|}+\big{|}\Psi(y,s)-\Psi^{*}(y/\sqrt{s})\big{|}\Big{)}\to
0\quad\textup{as}\;\;s\to+\infty,$ (3.1)

where $\Phi^{*}$ and $\Psi^{*}$ are the profiles defined in Theorem 1.1. From
the formal computation of an approximate blowup profile presented in the
previous section, we linearize (2.5) around $(\varphi,\psi)$ defined in (2.29)
instead of $(\Phi^{*},\Psi^{*})$, namely that we introduce

$\binom{\Lambda}{\Upsilon}=\binom{\Phi}{\Psi}-\binom{\varphi}{\psi},$ (3.2)

which leads the linearized system

$\partial_{s}\binom{\Lambda}{\Upsilon}=\Big{(}\mathcal{H}+Mc_{1}+V(y,s)\Big{)}\binom{\Lambda}{\Upsilon}+\binom{F_{1}(\Upsilon,y,s)}{F_{2}(\Lambda,y,s)}+\binom{R_{1}(y,s)}{R_{2}(y,s)},$
(3.3)

where $\mathcal{H}$ and $Mc_{1}$ are defined in (2.12),

$V(y,s)=\begin{pmatrix}0&p\big{(}\psi^{p-1}-\gamma^{p-1}\big{)}\\\
q\big{(}\varphi^{q-1}-\Gamma^{q-1}\big{)}&0\end{pmatrix}\equiv\begin{pmatrix}0&V_{1}\\\
V_{2}&0\end{pmatrix},$ (3.4)
$\binom{F_{1}(\Upsilon,y,s)}{F_{2}(\Lambda,y,s)}=\binom{|\Upsilon+\psi|^{p-1}(\Upsilon+\psi)-\psi^{p}-p\psi^{p-1}\Upsilon}{|\Lambda+\varphi|^{q-1}(\Lambda+\varphi)-\varphi^{q}-q\varphi^{q-1}\Lambda},$
(3.5)

and

$\binom{R_{1}(y,s)}{R_{2}(y,s)}=\binom{-\partial_{s}\varphi+\Delta\varphi-\frac{1}{2}y\cdot\nabla\varphi-\left(\frac{p+1}{pq-1}\right)\varphi+\psi^{p}}{-\partial_{s}\psi+\mu\Delta\psi-\frac{1}{2}y\cdot\nabla\psi-\left(\frac{q+1}{pq-1}\right)\psi+\varphi^{q}}.$
(3.6)

Our aim turns to construct for system (3.3) a global in time solution
$(\Lambda,\Upsilon)$ verifying

$\sup_{y\in\mathbb{R}^{N}}\Big{(}\big{|}\Lambda(y,s)\big{|}+\big{|}\Upsilon(y,s)\big{|}\Big{)}\to
0\quad\textup{as}\;\;s\to+\infty.$ (3.7)

Since the solution $(\Lambda,\Upsilon)$ goes to zero as $s\to+\infty$ and the
nonlinear term $(F_{1},F_{2})$ is built to be quadratic and the error term
$(R_{1},R_{2})$ is of the size $s^{-1}$, we see that the dynamics of (3.3) are
strongly influenced by the linear part $\mathcal{H}+Mc_{1}+V$. Here the
potential $V$ behaves differently as follows:
\- Outer region, i.e. $|y|\gtrsim\sqrt{s}$: for all $\epsilon>0$, there exists
$K_{\epsilon}>0$ and $s_{\epsilon}>0$ such that

$\sup_{|y|\geq K_{\epsilon}\sqrt{s},s\geq s_{\epsilon}}|V(y,s)|\leq\epsilon.$

From Lemma 2.1, we see that the linear operator $\mathcal{H}+Mc_{1}+V$ behaves
as one with fully negative spectrum in the outer region, which makes analysis
in this region simpler.
\- Inner region, i.e. $|y|\lesssim\sqrt{s}$: the potential $V$ is considered
as a perturbation of the linear part $\mathcal{H}+Mc_{1}$.

Since the behavior of $V$ in the inner and outer regions is different, this
suggests to consider the dynamics of (3.3) for $|y|\lesssim\sqrt{s}$ and
$|y|\gtrsim\sqrt{s}$ separately. To this end, we introduce the cut-off
function

$\chi(y,s)=\chi_{0}\left(\frac{|y|}{K\sqrt{s}}\right),\quad\chi_{0}\in\mathcal{C}_{0}^{\infty}(\mathbb{R}^{+},[0,1]),\quad\chi_{0}(r)=\left\\{\begin{array}[]{ll}1&\textup{for}\;\;r\in[0,1],\\\
0&\textup{for}\;\;r\geq 2,\end{array}\right.$ (3.8)

where $K$ is a positive constant to be fixed large enough. We then define

$\binom{\Lambda_{e}}{\Upsilon_{e}}=\big{(}1-\chi(y,s)\big{)}\binom{\Lambda}{\Upsilon}$
(3.9)

and consider the decomposition

$\binom{\Lambda}{\Upsilon}(y,s)=\sum_{n\leq
M}\left[\theta_{n}(s)\binom{f_{n}}{g_{n}}+\tilde{\theta}_{n}\binom{\tilde{f}_{n}}{\tilde{g}_{n}}\right]+\binom{\Lambda_{-}}{\Upsilon_{-}}(y,s),$
(3.10)

where $\theta_{n}=\Pi_{n}\binom{\Lambda}{\Upsilon}$ and
$\tilde{\theta}_{n}=\tilde{\Pi}_{n}\binom{\Lambda}{\Upsilon}$ with $\Pi_{n}$
and $\tilde{\Pi}_{n}$ being the projections onto the modes
$\binom{f_{n}}{g_{n}}$ and $\binom{\tilde{f}_{n}}{\tilde{g}_{n}}$
respectively, and
$\binom{\Lambda_{-}}{\Upsilon_{-}}=\Pi_{-,M}\binom{\Lambda}{\Upsilon}$ is
called the infinite-dimensional part with $\Pi_{-,M}$ being the projector on
the eigen-subspace corresponding the spectrum of $\mathcal{H}$ lower than
$\frac{1-M}{2}$. Note that the decomposition 3.10 is unique.

#### Preparation of initial data and Definition of the shrinking set:

Given $A>1$ and $s_{0}\geq e$, we consider the initial data for system (3.3)
of the form

$\binom{\Lambda}{\Upsilon}_{A,s_{0},d_{0},d_{1}}(y)=\frac{A}{s_{0}^{2}}\left[d_{0}\binom{f_{0}}{g_{0}}+d_{1}\cdot\binom{f_{1}}{g_{1}}\right]\chi(y,s_{0}),$
(3.11)

where $d_{0}\in\mathbb{R}$ and $d_{1}\in\mathbb{R}^{N}$ are parameters of the
problem. Our aim is to show that for a fixed large constant $A$, then
$s_{0}=s_{0}(A)$ is fixed large as well, there exist
$(d_{0},d_{1})\in\mathbb{R}^{1+N}$ so that system (3.3) with initial data at
$s=s_{0}$ given by (3.11) has the unique solution $(\Lambda,\Upsilon)$
satisfies (3.7). More precisely, we will show that the solution
$(\Lambda,\Upsilon)$ belongs to the following shrinking set:

######  Definition 3.1 (Definition of a shrinking set).

For all $A\geq 1$ and $s\geq e$, we defined $\mathcal{V}_{A}(s)$ as the set of
all $(\Lambda,\Upsilon)\in L^{\infty}(\mathbb{R}^{N})\times
L^{\infty}(\mathbb{R}^{N})$ such that

$|\theta_{0}(s)|\leq\frac{A}{s^{2}},\quad|\theta_{1}(s)|\leq\frac{A}{s^{2}},\quad|\theta_{2}(s)|\leq\frac{A^{4}\log
s}{s^{2}},$
$|\theta_{j}(s)|\leq\frac{A^{j}}{s^{\frac{j+1}{2}}},\quad|\tilde{\theta}_{j}(s)|\leq\frac{A^{j}}{s^{\frac{j+1}{2}}}\;\;\text{for}\;\;3\leq
j\leq
M,\quad|\tilde{\theta}_{i}(s)|\leq\frac{A^{2}}{s^{2}}\;\;\text{for}\;\;i=0,1,2,$
$\left\|\frac{\Lambda_{-}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{A^{M+1}}{s^{\frac{M+2}{2}}},\quad\left\|\frac{\Upsilon_{-}(y,s)}{1+|y|^{M+1}}\right\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{A^{M+1}}{s^{\frac{M+2}{2}}},$
$\|\Lambda_{e}(s)\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{A^{M+2}}{\sqrt{s}},\quad\|\Upsilon_{e}(s)\|_{L^{\infty}(\mathbb{R}^{N})}\leq\frac{A^{M+2}}{\sqrt{s}},$

where $\Lambda_{e},\Upsilon_{e}$ are defined by (3.9),
$\Lambda_{-},\Upsilon_{-}$, $\theta_{n}$, $\tilde{\theta}_{n}$ are defined as
in decomposition (3.10).

######  Remark 3.2.

We can check that if $\binom{\Lambda}{\Upsilon}\in\mathcal{V}_{A}(s)$ for
$s\geq e$, then

$\|\Lambda(s)\|_{L^{\infty}(\mathbb{R})}+\|\Upsilon(s)\|_{L^{\infty}(\mathbb{R})}\leq\frac{CA^{M+2}}{\sqrt{s}},$
(3.12)

for some positive constant $C$, hence, estimate (3.7) is proved.

In the following we make sure that the initial data (3.11) belongs to
$\mathcal{V}_{A}(s_{0})$.

######  Proposition 3.3 (Properties of initial data (3.11)).

For each $A\gg 1$, there exist $s_{0}(A)\gg 1$ and a cuboid
$\mathcal{D}_{s_{0}}\subset[-A,A]^{1+N}$ such that for all
$(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$, the following properties hold:

  * (i)

The initial data (3.11) belongs to $\mathcal{V}_{A}(s_{0})$ with strict
inequalities except for the estimates of $\theta_{0}(s_{0})$ and
$\theta_{1}(s_{0})$.

  * (ii)

The map $\Theta:\mathcal{D}_{s_{0}}\to\mathbb{R}^{1+N}$, defined as
$\Theta(d_{0},d_{1})=(\theta_{0}(s_{0}),\theta_{1}(s_{0}))$, is linear, one to
one from $\mathcal{D}_{s_{0}}$ to $[-As_{0}^{-2},As_{0}^{-2}]^{1+N}$, and maps
$\partial\mathcal{D}_{s_{0}}$ into
$\partial\big{(}[-As_{0}^{-2},As_{0}^{-2}]^{1+N}\big{)}$. Moreover, the degree
of $\Theta$ on the boundary is different from zero.

###### Proof.

See Proposition 3.3 in [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag]. ∎

#### Existence of a solution to (3.3) trapped in $\mathcal{V}_{A}(s)$:

From Remark 3.2, we aim at proving the following.

######  Proposition 3.4 (Existence of a solution of (3.3) trapped in
$\mathcal{V}_{A}(s)$).

There exists $A_{1}$ such that for all $A\geq A_{1}$, there exists
$s_{0,1}(A)$ such that for all $s_{0}\geq s_{0,1}$, there exists
$(d_{0},d_{1})$ such that if $\binom{\Lambda}{\Upsilon}$ is the solution of
(3.3) with initial data at $s_{0}$ given by (3.11), then
$\binom{\Lambda(s)}{\Upsilon(s)}\in\mathcal{V}_{A}(s)$ for all $s\geq s_{0}$.

###### Proof.

For a fixed constant $A\gg 1$ and $s_{0}(A)\gg 1$, we note from the local
Cauchy problem for system (1.1)-(1.2) in $L^{\infty}(\mathbb{R}^{N})\times
L^{\infty}(\mathbb{R}^{N})$ that for each initial data (3.11), system (3.3)
has a unique solution which stays in $\mathcal{V}_{A}(s)$ until some maximum
time $s_{*}=s_{*}(d_{0},d_{1})$. If $s_{*}(d_{0},d_{1})=+\infty$ for some
$(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$, then the proof is complete. Otherwise,
we argue by contradiction and suppose that $s_{*}(d_{0},d_{1})<+\infty$ for
any $(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$. By continuity and the definition of
$s_{*}$, we note that the solution at time $s_{*}$ is on the boundary of
$\mathcal{V}_{A}(s_{*})$. Thus, at least one of the inequalities in the
definition of $\mathcal{V}_{A}(s_{*})$ is an equality. In the following
proposition, we show that this can happen only for the two components
$\theta_{0}(s_{*})$ and $\theta_{1}(s_{*})$.

######  Proposition 3.5 (Reduction to a finite dimensional problem).

Assume that $\binom{\Lambda}{\Upsilon}$ is a solution of (3.3) with initial
data at $s=s_{0}$ given by (3.11) with $(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$,
and $\binom{\Lambda(s)}{\Upsilon(s)}\in\mathcal{V}_{A}(s)$ for all
$s\in[s_{0},s_{1}]$ for some $s_{1}\geq s_{0}$ and
$\binom{\Lambda(s_{1})}{\Upsilon(s_{1})}\in\partial\mathcal{V}_{A}(s_{1})$,
then

  * (i)

$\big{(}\theta_{0}(s_{1}),\theta_{1}(s_{1})\big{)}\in\partial\left(\left[-\frac{A}{s_{1}^{2}},\frac{A}{s_{1}^{2}}\right]\right)^{1+N}$.

  * (ii)

There exists $\nu_{0}>0$ such that

$\forall\nu\in(0,\nu_{0}),\quad\binom{\Lambda(s_{1}+\nu)}{\Upsilon(s_{1}+\nu)}\not\in\mathcal{V}_{A}(s_{1}+\nu).$

###### Proof.

The proof of Proposition 3.5 is a direct consequence of the dynamics of system
(3.3). The idea is to project system (3.3) on the different components of the
decomposition (3.10) and (3.9). For all details of the proof, see Section 5.2
in [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag]. ∎

From part $(i)$ of Proposition 3.5, we see that

$\big{(}\theta_{0}(s_{*}),\theta_{1}(s_{*})\big{)}\in\partial\left(\left[-\frac{A}{s_{*}^{2}},\frac{A}{s_{*}^{2}}\right]\right)^{1+N}.$

Hence, we may define the rescaled flow $\Theta$ at $s=s_{*}$ as follows:

$\displaystyle\Theta:\mathcal{D}_{s_{0}}$
$\displaystyle\to\partial\big{(}[-1,1]^{1+N}\big{)}$
$\displaystyle(d_{0},d_{1})$
$\displaystyle\mapsto\frac{s_{*}^{2}}{A}\big{(}\theta_{0},\theta_{1}\big{)}_{d_{0},d_{1}}(s_{*}),$

which is continuous from part $(ii)$ of Proposition 3.5. On the other hand,
from Proposition 3.3, we have the strict inequalities for the other components
for $(d_{0},d_{1})\in\partial\mathcal{D}_{s_{0}}$. Applying part $(ii)$ of
Proposition 3.5, we see that $\binom{\Lambda(s)}{\Upsilon(s)}$ must leave
$\mathcal{V}_{A}(s)$ at $s=s_{0}$, hence, $s_{*}(d_{0},d_{1})=s_{0}$.
Recalling from part $(ii)$ of Proposition 3.3 that the degree of $\Theta$ on
the boundary is different from zero. A contradiction then follows from the
index theory. This concludes that there must exist
$(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$ such that for all $s\geq s_{0}$,
$\binom{\Lambda(s)}{\Upsilon(s)}\in\mathcal{V}_{A}(s)$. This concludes the
proof of Proposition 3.4 as well as part $(ii)$ of Theorem 1.1. ∎

#### Equivalence of the final blowup profile:

We present the main argument for the proof of part $(iii)$ of Theorem 1.1. For
each $x_{0}\neq 0$ with $|x_{0}|\ll 1$, we introduce for all
$(\xi,\tau)\in\mathbb{R}\times\left[-\frac{t_{0}(x_{0})}{T-t_{0}(x_{0})},1\right)$
the auxillary functions

$g(x_{0},\xi,\tau)=(T-t_{0}(x_{0}))^{\alpha}u(x,t),\quad
h(x_{0},\xi,\tau)=(T-t_{0}(x_{0}))^{\beta}v(x,t),$

where

$x=x_{0}+\xi\sqrt{T-t_{0}(x_{0})},\quad t=t_{0}(x_{0})+\tau(T-t_{0}(x_{0})),$
(3.13)

and $t_{0}(x_{0})$ is uniquely determined by

$|x_{0}|=K\sqrt{(T-t_{0}(x_{0}))|\log(T-t_{0}(x_{0}))|}\quad\textup{for a
fixed constant $K\gg 1$.}$ (3.14)

From the invariance of system (1.1)-(1.2) under the scaling,
$(g(x_{0},\xi,\tau),h(x_{0},\xi,\tau))$ also satisfies (1.1)-(1.2). From
(3.13), (3.13) and the asymptotic behavior (1.6), we have

$\sup_{|\xi|\leq
2|\log(T-t_{0}(x_{0}))|^{1/4}}\left|g(x_{0},\xi,0)-\Phi^{*}(K)\right|\leq\frac{C}{|\log(T-t_{0}(x_{0}))|^{1/4}}\to
0,$

and

$\sup_{|\xi|\leq
2|\log(T-t_{0}(x_{0}))|^{1/4}}\left|h(x_{0},\xi,0)-\Psi^{*}(K)\right|\leq\frac{C}{|\log(T-t_{0}(x_{0}))|^{1/4}}\to
0,$

as $|x_{0}|\to 0$. From the continuity with respect to initial data for system
(1.1)-(1.2) associated to a space-localization in the ball
$B(0,|\xi|<|\log(T-t_{0}(x_{0}))|^{1/4})$, we can show that

$\sup_{|\xi|\leq
2|\log(T-t_{0}(x_{0}))|^{1/4},0\leq\tau<1}\left|g(x_{0},\xi,0)-\hat{g}_{K}(\tau)\right|\leq\epsilon(x_{0})\to
0,$

and

$\sup_{|\xi|\leq
2|\log(T-t_{0}(x_{0}))|^{1/4},0\leq\tau<1}\left|h(x_{0},\xi,0)-\hat{h}_{K}(\tau)\right|\leq\epsilon(x_{0})\to
0,$

as $x_{0}\to 0$, where

$\hat{g}_{K}(\tau)=\Gamma(1-\tau+bK^{2})^{-\alpha},\quad\hat{h}_{K}(\tau)=\gamma(1-\tau+bK^{2})^{-\beta},$

is the solution of system (1.1)-(1.2) with constant initial data
$(\Phi^{*}(K),\Psi^{*}(K))$.

Making $\tau\to 1$ and using (3.13) yields

$\displaystyle u^{*}(x_{0})$ $\displaystyle=\lim_{t\to
T}u(x,t)=(T-t_{0}(x_{0}))^{-\alpha}\lim_{\tau\to
1}g(x_{0},0,\tau)\sim(T-t_{0}(x_{0}))^{-\alpha}\hat{g}_{K}(1),$ $\displaystyle
v^{*}(x_{0})$ $\displaystyle=\lim_{t\to
T}v(x,t)=(T-t_{0}(x_{0}))^{-\beta}\lim_{\tau\to
1}h(x_{0},0,\tau)\sim(T-t_{0}(x_{0}))^{-\beta}\hat{h}_{K}(1),$

as $|x_{0}|\to 0$. Using the relation (3.14), we obtain

$|\log(T-t_{0}(x_{0}))|\sim 2\log|x_{0}|,\quad
T-t_{0}(x_{0})\sim\frac{|x_{0}|^{2}}{2K^{2}|\log|x_{0}||}\quad\text{as}\quad|x_{0}|\to
0,$

hence,

$u^{*}(x_{0})\sim\Gamma\left(\frac{b|x_{0}|^{2}}{2|\log|x_{0}||}\right)^{-\alpha},\quad
v^{*}(x_{0})\sim\gamma\left(\frac{b|x_{0}|^{2}}{2|\log|x_{0}||}\right)^{-\beta},$

as $|x_{0}|\to 0$. This concludes the proof of part $(iii)$ of Theorem 1.1.
Note that part $(iii)$ directly gives the single point blowup which is the
conclusion of part $(i)$. This completes the proof of Theorem 1.1. For the
proof of Theorem 1.4, we refer to [Ghoul et al.(2018c)Ghoul, Nguyen, and
Zaag].

###  3.2. The exponential case (1.3).

In this section we shall sketch those variants of the previous arguments which
are required for the proof of Theorem 1.4. All details of the proof can be
found in [Ghoul et al.(2018d)Ghoul, Nguyen, and Zaag]. The main difference
between the two cases is the presence of the nonlinear gradient terms
$\frac{|\nabla\Phi|^{2}}{\Phi}$ and $\frac{|\nabla\Psi|^{2}}{\Psi}$ in (2.8)
after making the change of variables (2.2). In view of the approximate profile
(2.32), the control of these terms is delicate, in particular when the
solution goes to zero in the intermediate zone. In order to treat them, we
introduce a very careful control of the solution in a 3-fold shrinking set
defined as follows: For $K_{0}>0$, $\epsilon_{0}>0$ and $t\in[0,T)$, we set

$\displaystyle\mathcal{D}_{1}(t)$ $\displaystyle=\left\\{x\;\Big{|}\;|x|\leq
K_{0}\sqrt{|\ln(T-t)|(T-t)}\right\\}$
$\displaystyle\quad\equiv\left\\{x\;\big{|}\;|y|\leq
K_{0}\sqrt{s}\right\\}\equiv\left\\{x\;\Big{|}\;|z|\leq K_{0}\right\\},$
$\displaystyle\mathcal{D}_{2}(t)$
$\displaystyle=\left\\{x\;\Big{|}\;\frac{K_{0}}{4}\sqrt{|\ln(T-t)|(T-t)}\leq|x|\leq\epsilon_{0}\right\\}$
$\displaystyle\quad\equiv\left\\{x\;\Big{|}\;\frac{K_{0}}{4}\sqrt{s}\leq|y|\leq\epsilon_{0}e^{\frac{s}{2}}\right\\}\equiv\left\\{x\;\Big{|}\;\frac{K_{0}}{4}\leq|z|\leq\frac{\epsilon_{0}}{\sqrt{s}}e^{\frac{s}{2}}\right\\},$
$\displaystyle\mathcal{D}_{3}(t)$
$\displaystyle=\left\\{x\;\Big{|}\;|x|\geq\frac{\epsilon_{0}}{4}\right\\}\equiv\left\\{x\;\big{|}\;|y|\geq\frac{\epsilon_{0}}{4}e^{\frac{s}{2}}\right\\}\equiv\left\\{x\;\Big{|}\;|z|\geq\frac{\epsilon_{0}}{4\sqrt{s}}e^{\frac{s}{2}}\right\\}.$

\- In the blowup region $\mathcal{D}_{1}$, we linearize (2.21) around the
approximate profile (2.32), namely that
$(\Lambda,\Upsilon)=(\Phi,\Psi)-(\varphi,\psi)$ solves the system

$\partial_{s}\binom{\Lambda}{\Upsilon}=\Big{(}\mathcal{H}+Mc_{2}+V(y,s)\Big{)}\binom{\Lambda}{\Upsilon}+\binom{q}{p}\Lambda\Upsilon+\binom{R_{1}}{R_{2}}+\binom{G_{1}}{G_{2}},$
(3.15)

where $\mathcal{H}$ and $Mc_{2}$ are defined by (2.12),

$V(y,s)=\begin{pmatrix}q\psi-1&\quad q\big{(}\phi-1/p\big{)}\\\
p\big{(}\psi-1/q\big{)}&\quad
p\phi-1\end{pmatrix}=\begin{pmatrix}V_{1}&V_{2}\\\ V_{3}&V_{4}\end{pmatrix},$
(3.16)
$\binom{G_{1}}{G_{2}}=\binom{-|\nabla(\Lambda+\phi)|^{2}(\Lambda+\phi)^{-1}+|\nabla\phi|^{2}\phi^{-1}}{-\mu|\nabla(\Upsilon+\psi)|^{2}(\Upsilon+\psi)^{-1}+\mu|\nabla\psi|^{2}\psi^{-1}},$
(3.17)

and

$\binom{R_{1}}{R_{2}}=\binom{-\partial_{s}\phi+\Delta\phi-\frac{1}{2}y\cdot\nabla\phi-\phi+q\phi\psi-|\nabla\phi|^{2}\phi^{-1}}{-\partial_{s}\psi+\mu\Delta\psi-\frac{1}{2}y\cdot\nabla\psi-\psi+p\phi\psi-\mu|\nabla\psi|^{2}\psi^{-1}}.$
(3.18)

The analysis is similar as for the polynomial case according to the
decomposition (3.10) and the definition (3.9).

\- In the intermediate region $\mathcal{D}_{2}$, we control $(u,v)$ by
introducing the following auxillary functions $(\tilde{u},\tilde{v})$ defined
for $x\neq 0$,

$\left\\{\begin{array}[]{l}\tilde{u}(x,\xi,\tau)=\frac{1}{q}\ln\sigma(x)+u\Big{(}x+\xi\sqrt{\sigma(x)},t(x)+\tau\sigma(x)\Big{)},\\\
\tilde{v}(x,\xi,\tau)=\frac{1}{p}\ln\sigma(x)+v\Big{(}x+\xi\sqrt{\sigma(x)},t(x)+\tau\sigma(x)\Big{)},\end{array}\right.$
(3.19)

where $t(x)$ is uniquely defined for $|x|$ sufficiently small by

$|x|=\frac{K_{0}}{4}\sqrt{\sigma(x)|\ln\sigma(x)|}\quad\text{with}\quad\sigma(x)=T-t(x).$
(3.20)

By the scaling invariance of the problem, we see that $(\tilde{u},\tilde{v})$
also satisfies system (1.1)-(1.3). We prove that $(\tilde{u},\tilde{v})$
behaves for

$|\xi|\leq\alpha_{0}\sqrt{|\ln\sigma(x)|}\quad\text{and}\quad\tau\in\left[\frac{t_{0}-t(x)}{\sigma(x)},1\right)$

for some $t_{0}<T$ and $\alpha_{0}>0$, like the solution of the ordinary
differential system

$\partial_{\tau}\hat{u}=e^{p\hat{v}},\quad\partial_{\tau}\hat{v}=e^{q\hat{u}},$
(3.21)

subject to the initial data

$\hat{u}(0)=-\frac{1}{q}\ln\left[p\left(1+\frac{K_{0}^{2}/16}{2(\mu+1)}\right)\right],\quad\hat{v}(0)=-\frac{1}{p}\ln\left[q\left(1+\frac{K_{0}^{2}/16}{2(\mu+1)}\right)\right].$

The explicit solution is given by

$\hat{u}(\tau)=-\frac{1}{q}\ln\left[p\left(1-\tau+\frac{K_{0}^{2}/16}{2(\mu+1)}\right)\right],\quad\hat{v}(\tau)=-\frac{1}{p}\ln\left[q\left(1-\tau+\frac{K_{0}^{2}/16}{2(\mu+1)}\right)\right].$
(3.22)

The analysis in $\mathcal{D}_{2}$ directly yields the conclusion of part
$(iii)$ of Theorem 1.4.

\- In $\mathcal{D}_{3}$, we directly control $(u,v)$ by using the local in
time well-posedness of the Cauchy problem for system (1.1).

The following definition of the shrinking set to trap the solution is the
crucial difference in comparison with the existence proof for the polynomial
case.

######  Definition 3.6 (Definition of a shrinking set).

For all $t_{0}<T$, $K_{0}>0$, $\epsilon_{0}>0$, $\alpha_{0}>0$, $A>0$,
$\delta_{0}>0$, $\eta_{0}>0$, $C_{0}>0$, for all $t\in[t_{0},T)$, we define
$\;\mathcal{S}(t_{0},K_{0},\epsilon_{0},\alpha_{0},A,\delta_{0},\eta_{0},C_{0},t)$
being the set of all functions $(u,v)$ such that

  * (i)

(Control in $\mathcal{D}_{1}$)
$\quad\binom{\Lambda(s)}{\Upsilon(s)}\in\mathcal{V}_{A}(s)$, where
$\mathcal{V}_{A}(s)$ is introduced in Definition 3.1.

  * (ii)

(Control in $\mathcal{D}_{2}$) For all
$|x|\in\left[\frac{K_{0}}{4}\sqrt{|\ln(T-t)|(T-t)},\epsilon_{0}\right]$,
$\tau=\tau(x,t)=\frac{t-t(x)}{\sigma(x)}$ and
$|\xi|\leq\alpha_{0}\sqrt{\ln\sigma(x)}$,

$\displaystyle\left|\tilde{u}(x,\xi,\tau)-\hat{u}(\tau)\right|\leq\delta_{0},\quad|\nabla_{\xi}\tilde{u}(x,\xi,\tau)|\leq\frac{C_{0}}{\sqrt{|\ln\sigma(x)|}},$
$\displaystyle\left|\tilde{v}(x,\xi,\tau)-\hat{v}(\tau)\right|\leq\delta_{0},\quad|\nabla_{\xi}\tilde{v}(x,\xi,\tau)|\leq\frac{C_{0}}{\sqrt{|\ln\sigma(x)|}},$

where $\tilde{u},\tilde{v}$, $\hat{u}$, $\hat{v}$, $t(x)$ and $\sigma(x)$ are
defined in (3.19), (3.22) and (3.20) respectively.

  * (iii)

(Control in $\mathcal{D}_{3}$) For all $|x|\geq\frac{\epsilon_{0}}{4}$,

$\displaystyle|\nabla_{x}^{i}u(x,t)-\nabla_{x}^{i}u(x,t_{0})|\leq\eta_{0}\quad\text{and}\quad|\nabla_{x}^{i}v(x,t)-\nabla_{x}^{i}v(x,t_{0})|\leq\eta_{0}\quad\text{for}\;\;i=0,1.$

######  Remark 3.7.

In comparison with Definition 3.1, the shrinking set $\mathcal{S}$ has
additional estimates in the domains $\mathcal{D}_{2}$ and $\mathcal{D}_{3}$.
These estimates are crucially needed to achieve the control of the nonlinear
gradient term $\binom{G_{1}}{G_{2}}$ appearing in (3.15).

After defining the shrinking set $\mathcal{S}$ to trap the solution, we need a
suitable initial data for (3.15) so that the corresponding solution gradually
belongs to $\mathcal{S}(t)$ for all $t\in[t_{0},T)$. To this end, we consider
the following functions depending on $(N+1)$ parameters
$(d_{0},d_{1})\in\mathbb{R}^{1+N}$:

$\displaystyle\binom{qu}{pv}_{d_{0},d_{1}}(x,t_{0})$
$\displaystyle=\binom{\hat{u}_{*}(x)}{\hat{v}_{*}(x)}\Big{(}1-\chi_{1}(x,t_{0})\Big{)}+\left\\{\binom{1}{1}s_{0}+\ln\left[\binom{\phi}{\psi}(y_{0},s_{0})\right]\right\\}\chi_{1}(x,t_{0})$
$\displaystyle+\ln\left\\{\left(d_{0}\binom{f_{0}(y_{0})}{g_{0}(y_{0})}+d_{1}.\binom{f_{1}(y_{0})}{g_{1}(y_{0})}\right)\frac{A^{2}}{s_{0}^{2}}\chi(16y_{0},s_{0})\right\\}\chi_{1}(x,t_{0}),$
(3.23)

where $s_{0}=-\ln(T-t_{0})$, $y_{0}=xe^{\frac{s_{0}}{2}}$, $\phi$ and $\psi$
are defined by (2.32), $\binom{f_{0}}{g_{0}}$ and $\binom{f_{1}}{g_{1}}$ are
the eigenfunctions introduced in Lemma 2.1, $\chi$ is the cut-off function
defined by (3.8),

$\chi_{1}(x,t_{0})=\chi_{0}\left(\frac{|x|}{|\ln(T-t_{0})|\sqrt{T-t_{0}}}\right)=\chi_{0}\left(\frac{y_{0}}{s_{0}}\right),$

and $(\hat{u}_{*},\hat{v}_{*})\in\mathcal{C}^{\infty}(\mathbb{R}^{N}\
\\{0\\})\times\mathcal{C}^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ is defined
by

$\hat{u}_{*}(x)=\left\\{\begin{array}[]{ll}\ln\left(\frac{4(\mu+1)|\ln|x||}{p|x|^{2}}\right)&\quad\text{for}\quad|x|\leq
C(a),\\\ -\ln\left(1+a|x|^{2}\right)&\quad\text{for}\quad|x|\geq
1,\end{array}\right.$
$\hat{v}_{*}(x)=\left\\{\begin{array}[]{ll}\ln\left(\frac{4(\mu+1)|\ln|x||}{q|x|^{2}}\right)&\quad\text{for}\quad|x|\leq
C(a),\\\ -\ln\left(1+a|x|^{2}\right)&\quad\text{for}\quad|x|\geq
1.\end{array}\right.$

After having a proper definition of initial data and the shrinking set, the
remaining step is to show that there exists $(d_{0},d_{1})\in\mathbb{R}^{1+N}$
such that system (1.1)-(1.3) with initial data (3.23) has a unique solution
$(u,v)\in\mathcal{S}(t)$ for all $t\in[t_{0},T)$. The main argument of this
step is exactly the same as for the polynomial case, i.e. the proof of
Proposition 3.4. We refer the interested reader on the reduction to a finite
dimensional problem to Section 4 in [Ghoul et al.(2018d)Ghoul, Nguyen, and
Zaag] for all details. This concludes the proof of Theorem 1.4.

## References

  * [Andreucci et al.(1997)Andreucci, Herrero, and Velázquez] D. Andreucci, M. A. Herrero, and J. J. L. Velázquez.  Liouville theorems and blow up behaviour in semilinear reaction diffusion systems.  _Ann. Inst. H. Poincaré Anal. Non Linéaire_ , 14(1):1–53, 1997.  ISSN 0294-1449.  10.1016/S0294-1449(97)80148-5.  URL http://dx.doi.org/10.1016/S0294-1449(97)80148-5.
  * [Bressan(1990)] A. Bressan.  On the asymptotic shape of blow-up.  _Indiana Univ. Math. J._ , 39(4):947–960, 1990\.  ISSN 0022-2518.  10.1512/iumj.1990.39.39045.  URL http://dx.doi.org/10.1512/iumj.1990.39.39045.
  * [Bressan(1992)] A. Bressan.  Stable blow-up patterns.  _J. Differential Equations_ , 98(1):57–75, 1992\.  ISSN 0022-0396.  10.1016/0022-0396(92)90104-U.  URL http://dx.doi.org/10.1016/0022-0396(92)90104-U.
  * [Bricmont and Kupiainen(1994)] J. Bricmont and A. Kupiainen.  Universality in blow-up for nonlinear heat equations.  _Nonlinearity_ , 7(2):539–575, 1994.  ISSN 0951-7715.  URL http://stacks.iop.org/0951-7715/7/539.
  * [C.(2016a)] Collot C.  Non radial type II blow up for the energy supercritical semilinear heat equation.  _arXiv:1604.02856_ , 2016a.  URL http://arxiv.org/abs/1604.02856.
  * [C.(2016b)] Collot C.  Type II blow up manifolds for the energy supercritical wave equation.  _arXiv:1407.4525_ , 2016b.  URL http://arxiv.org/abs/1407.4525.
  * [Caristi and Mitidieri(1994)] G. Caristi and E. Mitidieri.  Blow-up estimates of positive solutions of a parabolic system.  _J. Differential Equations_ , 113(2):265–271, 1994.  ISSN 0022-0396.  10.1006/jdeq.1994.1124.  URL http://dx.doi.org/10.1006/jdeq.1994.1124.
  * [Deng(1996)] K. Deng.  Blow-up rates for parabolic systems.  _Z. Angew. Math. Phys._ , 47(1):132–143, 1996\.  ISSN 0044-2275.  10.1007/BF00917578.  URL http://dx.doi.org/10.1007/BF00917578.
  * [Duong et al.(2018)Duong, Nguyen, and Zaag] G. K. Duong, V. T. Nguyen, and H. Zaag.  Construction of a stable blowup solution with a prescribed behavior for a non-scaling invariant semilinear heat equation.  _Tunisian Journal of Mathematics (to appear)_ , 2018.  URL https://arxiv.org/abs/1704.08580.
  * [Duyckaerts et al.(2013)Duyckaerts, Kenig, and Merle] T. Duyckaerts, C. Kenig, and F. Merle.  Classification of radial solutions of the focusing, energy-critical wave equation.  _Camb. J. Math._ , 1(1):75–144, 2013.  ISSN 2168-0930.  10.4310/CJM.2013.v1.n1.a3.  URL http://dx.doi.org/10.4310/CJM.2013.v1.n1.a3.
  * [Escobedo and Herrero(1991a)] M. Escobedo and M. A. Herrero.  Boundedness and blow up for a semilinear reaction-diffusion system.  _J. Differential Equations_ , 89(1):176–202, 1991a.  ISSN 0022-0396.  10.1016/0022-0396(91)90118-S.  URL http://dx.doi.org/10.1016/0022-0396(91)90118-S.
  * [Escobedo and Herrero(1991b)] M. Escobedo and M. A. Herrero.  A uniqueness result for a semilinear reaction-diffusion system.  _Proc. Amer. Math. Soc._ , 112(1):175–185, 1991b.  ISSN 0002-9939.  10.2307/2048495.  URL http://dx.doi.org/10.2307/2048495.
  * [Escobedo and Herrero(1993)] M. Escobedo and M. A. Herrero.  A semilinear parabolic system in a bounded domain.  _Ann. Mat. Pura Appl. (4)_ , 165:315–336, 1993.  ISSN 0003-4622.  10.1007/BF01765854.  URL http://dx.doi.org/10.1007/BF01765854.
  * [Fila and Souplet(2001)] M. Fila and P. Souplet.  The blow-up rate for semilinear parabolic problems on general domains.  _NoDEA Nonlinear Differential Equations Appl._ , 8(4):473–480, 2001.  ISSN 1021-9722.  10.1007/PL00001459.  URL http://dx.doi.org/10.1007/PL00001459.
  * [Friedman and Giga(1987)] A. Friedman and Y. Giga.  A single point blow-up for solutions of semilinear parabolic systems.  _J. Fac. Sci. Univ. Tokyo Sect. IA Math._ , 34(1):65–79, 1987.  ISSN 0040-8980.
  * [Ghoul et al.(2017)Ghoul, Nguyen, and Zaag] T. Ghoul, V. T. Nguyen, and H. Zaag.  Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term.  _J. Differential Equations_ , 263(8):4517 – 4564, 2017.  ISSN 0022-0396.  http://dx.doi.org/10.1016/j.jde.2017.05.023.  URL http://www.sciencedirect.com/science/article/pii/S0022039617302838.
  * [Ghoul et al.(2018a)Ghoul, Ibrahim, and Nguyen] T. Ghoul, S. Ibrahim, and V. T. Nguyen.  On the stability of type II blowup for the 1-corotational energy supercritical harmonic heat flow.  _Anal. PDE (to appear)_ , 2018a.  URL https://arxiv.org/abs/1611.08877.
  * [Ghoul et al.(2018b)Ghoul, Ibrahim, and Nguyen] T. Ghoul, S. Ibrahim, and V. T. Nguyen.  Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps.  _J. Differential Equations_ , 265(7):2968–3047, 2018b.  ISSN 0022-0396.  10.1016/j.jde.2018.04.058.  URL https://doi.org/10.1016/j.jde.2018.04.058.
  * [Ghoul et al.(2018c)Ghoul, Nguyen, and Zaag] T.-E. Ghoul, V. T. Nguyen, and H. Zaag.  Construction and stability of blowup solutions for a non-variational semilinear parabolic system.  _Ann. Inst. H. Poincaré Anal. Non Linéaire_ , 2018c.  ISSN 0294-1449.  https://doi.org/10.1016/j.anihpc.2018.01.003.  URL http://www.sciencedirect.com/science/article/pii/S0294144918300076.
  * [Ghoul et al.(2018d)Ghoul, Nguyen, and Zaag] T.-E. Ghoul, V. T. Nguyen, and H. Zaag.  Blowup solutions for a reaction-diffusion system with exponential nonlinearities.  _J. Differential Equations_ , 264(12):7523–7579, 2018d.  ISSN 0022-0396.  10.1016/j.jde.2018.02.022.  URL https://doi.org/10.1016/j.jde.2018.02.022.
  * [Herrero and Velázquez(1992)] M. A. Herrero and J. J. L. Velázquez.  Generic behaviour of one-dimensional blow up patterns.  _Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)_ , 19(3):381–450, 1992.  ISSN 0391-173X.  URL http://www.numdam.org/item?id=ASNSP_1992_4_19_3_381_0.
  * [Hillairet and Raphaël(2012)] M. Hillairet and P. Raphaël.  Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation.  _Anal. PDE_ , 5(4):777–829, 2012.  ISSN 2157-5045.  10.2140/apde.2012.5.777.  URL http://dx.doi.org/10.2140/apde.2012.5.777.
  * [Mahmoudi et al.(2015)Mahmoudi, Souplet, and Tayachi] N. Mahmoudi, P. Souplet, and S. Tayachi.  Improved conditions for single-point blow-up in reaction-diffusion systems.  _J. Differential Equations_ , 259(5):1898–1932, 2015.  ISSN 0022-0396.  10.1016/j.jde.2015.03.024.  URL http://dx.doi.org/10.1016/j.jde.2015.03.024.
  * [Martel et al.(2014)Martel, Merle, and Raphaël] Y. Martel, F. Merle, and P. Raphaël.  Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton.  _Acta Math._ , 212(1):59–140, 2014.  ISSN 0001-5962.  10.1007/s11511-014-0109-2.  URL http://dx.doi.org/10.1007/s11511-014-0109-2.
  * [Martel et al.(2015a)Martel, Merle, and Raphaël] Y. Martel, F. Merle, and P. Raphaël.  Blow up for the critical gKdV equation III: exotic regimes.  _Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)_ , 14(2):575–631, 2015a.  ISSN 0391-173X.
  * [Martel et al.(2015b)Martel, Merle, and Raphaël] Y. Martel, F. Merle, and P. Raphaël.  Blow up for the critical gKdV equation. II: Minimal mass dynamics.  _J. Eur. Math. Soc. (JEMS)_ , 17(8):1855–1925, 2015b.  ISSN 1435-9855.  10.4171/JEMS/547.  URL http://dx.doi.org/10.4171/JEMS/547.
  * [Masmoudi and Zaag(2008)] N. Masmoudi and H. Zaag.  Blow-up profile for the complex Ginzburg-Landau equation.  _J. Funct. Anal._ , 255(7):1613–1666, 2008.  ISSN 0022-1236.  10.1016/j.jfa.2008.03.008.  URL http://dx.doi.org/10.1016/j.jfa.2008.03.008.
  * [Merle and Raphael(2003)] F. Merle and P. Raphael.  Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation.  _Geom. Funct. Anal._ , 13(3):591–642, 2003.  ISSN 1016-443X.  10.1007/s00039-003-0424-9.  URL http://dx.doi.org/10.1007/s00039-003-0424-9.
  * [Merle and Raphael(2004)] F. Merle and P. Raphael.  On universality of blow-up profile for $L^{2}$ critical nonlinear Schrödinger equation.  _Invent. Math._ , 156(3):565–672, 2004.  ISSN 0020-9910.  10.1007/s00222-003-0346-z.  URL http://dx.doi.org/10.1007/s00222-003-0346-z.
  * [Merle and Raphael(2005a)] F. Merle and P. Raphael.  The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation.  _Ann. of Math. (2)_ , 161(1):157–222, 2005a.  ISSN 0003-486X.  10.4007/annals.2005.161.157.  URL http://dx.doi.org/10.4007/annals.2005.161.157.
  * [Merle and Raphael(2005b)] F. Merle and P. Raphael.  Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation.  _Comm. Math. Phys._ , 253(3):675–704, 2005b.  ISSN 0010-3616.  10.1007/s00220-004-1198-0.  URL http://dx.doi.org/10.1007/s00220-004-1198-0.
  * [Merle and Zaag(1997)] F. Merle and H. Zaag.  Stability of the blow-up profile for equations of the type $u_{t}=\Delta u+|u|^{p-1}u$.  _Duke Math. J._ , 86(1):143–195, 1997.  ISSN 0012-7094.  10.1215/S0012-7094-97-08605-1.  URL http://dx.doi.org/10.1215/S0012-7094-97-08605-1.
  * [Merle et al.(2013)Merle, Raphaël, and Rodnianski] F. Merle, P. Raphaël, and I. Rodnianski.  Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem.  _Invent. Math._ , 193(2):249–365, 2013.  ISSN 0020-9910.  10.1007/s00222-012-0427-y.  URL http://dx.doi.org/10.1007/s00222-012-0427-y.
  * [Merle et al.(2015)Merle, Raphaël, and Rodnianski] F. Merle, P. Raphaël, and I. Rodnianski.  Type II blow up for the energy supercritical NLS.  _Camb. Jour. Math._ , 3(4):439 – 617, 2015.  10.4310/CJM.2015.v3.n4.a1.  URL http://dx.doi.org/10.4310/CJM.2015.v3.n4.a1.
  * [Nguyen and Zaag(2016)] V. T. Nguyen and H. Zaag.  Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations.  _Ann. Scuola Norm. Sup. Pisa Cl. Sci._ , 16(4):1275–1314, 2016.  10.2422/2036-2145.201412_001.  URL http://dx.doi.org/10.2422/2036-2145.201412_001.
  * [Nguyen and Zaag(2017)] V. T. Nguyen and H. Zaag.  Finite degrees of freedom for the refined blow-up profile for a semilinear heat equation.  _Ann. Scient. Éc. Norm. Sup._ , 50(4):1241–1282, 2017.  URL http://smf4.emath.fr/Publications/AnnalesENS/4_50/html/ens_ann-sc_50_1241-1282.php.
  * [Nouaili and Zaag(2015)] N. Nouaili and H. Zaag.  Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation.  _Comm. Partial Differential Equations_ , 40(7):1197–1217, 2015.  ISSN 0360-5302.  10.1080/03605302.2015.1018997.  URL http://dx.doi.org/10.1080/03605302.2015.1018997.
  * [Nouaili and Zaag(2018)] N. Nouaili and H. Zaag.  Construction of a blow-up solution for the Complex Ginzburg-Landau equation in some critical case.  _Arch. Ration. Mech. Anal., to appear_ , 2018.  URL https://arxiv.org/abs/1703.00081.
  * [Raphaël and Rodnianski(2012)] P. Raphaël and I. Rodnianski.  Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems.  _Publ. Math. Inst. Hautes Études Sci._ , pages 1–122, 2012.  ISSN 0073-8301.  10.1007/s10240-011-0037-z.  URL http://dx.doi.org/10.1007/s10240-011-0037-z.
  * [Raphaël and Schweyer(2013)] P. Raphaël and R. Schweyer.  Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow.  _Comm. Pure Appl. Math._ , 66(3):414–480, 2013\.  ISSN 0010-3640.  10.1002/cpa.21435.  URL http://dx.doi.org/10.1002/cpa.21435.
  * [Raphaël and Schweyer(2014a)] P. Raphaël and R. Schweyer.  Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow.  _Anal. PDE_ , 7(8):1713–1805, 2014a.  ISSN 2157-5045.  10.2140/apde.2014.7.1713.  URL http://dx.doi.org/10.2140/apde.2014.7.1713.
  * [Raphaël and Schweyer(2014b)] P. Raphaël and R. Schweyer.  On the stability of critical chemotactic aggregation.  _Math. Ann._ , 359(1-2):267–377, 2014b.  ISSN 0025-5831.  10.1007/s00208-013-1002-6.  URL http://dx.doi.org/10.1007/s00208-013-1002-6.
  * [Schweyer(2012)] R. Schweyer.  Type II blow-up for the four dimensional energy critical semi linear heat equation.  _J. Funct. Anal._ , 263(12):3922–3983, 2012.  ISSN 0022-1236.  10.1016/j.jfa.2012.09.015.  URL http://dx.doi.org/10.1016/j.jfa.2012.09.015.
  * [Souplet(2005)] P. Souplet.  The influence of gradient perturbations on blow-up asymptotics in semilinear parabolic problems: a survey.  In _Nonlinear elliptic and parabolic problems_ , volume 64 of _Progr. Nonlinear Differential Equations Appl._ , pages 473–495. Birkhäuser, Basel, 2005.  10.1007/3-7643-7385-7_28.  URL http://dx.doi.org/10.1007/3-7643-7385-7_28.
  * [Souplet and Tayachi(2016)] P. Souplet and S. Tayachi.  Single-point blow-up for parabolic systems with exponential nonlinearities and unequal diffusivities.  _Nonlinear Anal._ , 138:428–447, 2016.  ISSN 0362-546X.  10.1016/j.na.2016.01.008.  URL http://dx.doi.org/10.1016/j.na.2016.01.008.
  * [Tayachi and Zaag(2015)] S. Tayachi and H. Zaag.  Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term.  _Actes du Colloque EDP-Normandie, Le Havre, arXiv:1610.01289_ , 2015\.
  * [Tayachi and Zaag(2016)] S. Tayachi and H. Zaag.  Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term.  _arXiv:1506.08306_ , 2016.
  * [Zaag(1998)] H. Zaag.  Blow-up results for vector-valued nonlinear heat equations with no gradient structure.  _Ann. Inst. H. Poincaré Anal. Non Linéaire_ , 15(5):581–622, 1998.  ISSN 0294-1449.  10.1016/S0294-1449(98)80002-4.  URL http://dx.doi.org/10.1016/S0294-1449(98)80002-4.
  * [Zaag(2001)] H. Zaag.  A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure.  _Comm. Pure Appl. Math._ , 54(1):107–133, 2001\.  ISSN 0010-3640.  10.1002/1097-0312(200101)54:1¡107::AID-CPA5¿3.3.CO;2-L.  URL http://dx.doi.org/10.1002/1097-0312(200101)54:1<107::AID-CPA5>3.3.CO;2-L.
  * [Zheng et al.(2002)Zheng, Zhao, and Chen] S. Zheng, L. Zhao, and F. Chen.  Blow-up rates in a parabolic system of ignition model.  _Nonlinear Anal._ , 51(4, Ser. A: Theory Methods):663–672, 2002.  ISSN 0362-546X.  URL https://doi.org/10.1016/S0362-546X(01)00849-5.


* [AK16] Sylvy Anscombe and Franz-Viktor Kuhlmann.  Notes on extremal and tame valued fields.  The Journal of Symbolic Logic , Volume 81 , Issue 2 , pp. 400-416, 2016.
  * [Bas78] Serban A. Basarab.  Some model theoery for henselian valued fields.  Journal of Algebra, 1978.
  * [Bél99] Luc Bélair.  Types dans les corps valués munis d’applications coefficients.  Illinois J. Math. 43(2):410- 425, 1999.
  * [Del84] Pierre Deligne.  Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0.  Représentations des groupes réductifs sur un corps local, - Hermann Paris, 1984.
  * [DS03] Jan Denef and Hans Schoutens.  On the decidability of the existential theory of $\mathbb{F}_{p}[[t]]$.  Fields Institute Communications Volume 00, 0000, (2003).
  * [Ers65] Ju.L. Ershov.  On elementary theories of local fields.  Algebra i Logika 4, No. 2, 5-30, 1965.
  * [FF14] Laurent Fargues and Jean-Marc Fontaine.  Vector bundles on curves and $p$-adic Hodge theory.  Chapter 2 in "Automorphic forms and Galois representations", 2014.
  * [FF19] Laurent Fargues and Jean-Marc Fontaine.  Courbes et fibrés vectoriels en théorie de Hodge $p$-adique.  available at https://webusers.imj-prg.fr/~laurent.fargues/", 2019.
  * [Hod93] Wilfrid Hodges.  Model theory.  Cambridge University Press, 1993.
  * [Kar21] Konstantinos Kartas.  The asymptotic theory of $p$-adic fields.  arXiv:2105.03771[math.LO], 2021.
  * [Kaz86] D. Kazhdan.  Representations of groups over close local fields.  Journal d’ Analyse Mathématique volume 47, pages 175-179, 1986\.
  * [Ked17] Kiran S. Kedlaya.  On the algebraicity of generalized power series.  Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry volume 58, pages 499-527, 2017.
  * [Koc74] Simon Kochen.  The model theory of local fields.  ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), pp. 384-425. Lecture Notes in Math., Vol. 499, Springer, Berlin, 1975., 1974.
  * [Koe18] Jochen Koenigsmann.  Decidability in local and global fields.  Proc. Int. Cong. of Math. Rio de Janeiro, Vol. 1 (45-60), 2018.
  * [KR21] Franz-Viktor Kuhlmann and Anna Rzepka.  The valuation theory of deeply ramified fields and its connection with defect extensions.  arXiv: 1811.04396v3[math.AC], 2021.
  * [Kra56] M. Krasner.  Approximation des corps valués complets de caractéristique $p=0$ par ceux de caractéristique 0.  Colloque d’Algebre Superleure Bruxelles, 1956.
  * [KT18] Kiran S. Kedlaya and M Temkin.  Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves.  Proceedings of the American Mathematical Society,, 2018.
  * [Kuh16] Franz Victor Kuhlmann.  The algebra and model theory of tame valued fields.  Journal für die reine und angewandte Mathematik, 2016.
  * [Lee20] Junguk Lee.  Hyperfields, truncated DVRs, and valued fields.  Journal of Number Theory, Volume 212, July 2020, Pages 40-71, 2020\.
  * [Lis21] Victor Lisinski.  Decidability of equal characteristic tame Hahn fields in the language $\textsc{L}_{t}$.  arXiv:2108.04132[math.LO], 2021.
  * [LL21] Junguk Lee and Wan Lee.  On the structure of certain valued fields.  Annals of Pure and Applied Logic Volume 172, Issue 4, 2021.
  * [Lur18] Jacob Lurie.  The Fargues-Fontaine curve.  Online notes available at https://www.math.ias.edu/~lurie/205.html, 2018.
  * [Mac86] Angus Macintyre.  Twenty years of $p$-adic model theory.  Logic Colloquium ’84 Elsevier Science Publishers B. K (North-Holland), 1986.
  * [Mar02] David Marker.  Model Theory : An Introduction.  Springer-Verlag New York, 2002.
  * [MB11] Laurent Moret-Bailly.  An extension of Greenberg′s theorem to general valuation rings.  Manuscripta mathematica, 2011.
  * [Mor17] Matthew Morrow.  The Fargues-Fontaine curve and diamonds.  Séminaire Bourbaki, 2017.
  * [RZ60] Abraham Robinson and Elias Zakon.  Elementary properties of ordered abelian groups.  Transactions of the American Mathematical Society, 1960.
  * [Sca99] Thomas Scanlon.  Quantifier elimination for the relative Frobenius.  Valuation Theory and Its Applications Volume II, Conference proceedings of the International Conference on Valuation Theory (Saskatoon), 1999.
  * [Sch] Peter Scholze.  How many untilts?  Mathoverflow https://mathoverflow.net/q/390522.
  * [Sch12] Peter Scholze.  Perfectoid spaces.  Publ. Math. Inst. Hautes Études Sei. 116, November 2012.
  * [Sch14] Peter Scholze.  Perfectoid spaces and their applications.  Proceedings of the ICM, 2014.
  * [Ser80] Jean-Pierre Serre.  Local Fields.  New York: Springer-Verlag, 1980.
  * [vdD99] Lou van den Dries.  On the elementary theory of rings of Witt vectors with a multiplicative set of representatives for the residue field.  Manuscripta Mathematica, 98(2), 1999.
  * [vdD12] Lou van den Dries.  Lectures on the Model Theory of Valued Fields.  Chapter 5 in the book "Model Theory in Algebra, Analysis and Arithmetic", Cetraro, Italy, 2012.
  * [Was97] Lawrence C. Washington.  Introduction to Cyclotomic Fields.  Springer Science & Business Media, Graduate texts in mathematics, 1997\.
  * [Zie72] Martin Ziegler.  Die elementare Theorie der henselschen Körper.  Dissertation thesis, 1972.

Mathematical Institute, Woodstock Road, Oxford OX2 6GG.

E-mail address<EMAIL_ADDRESS>