#  Refined blow-up asymptotics for a perturbed nonlinear heat equation with a
gradient and a non-local term

Bouthaina Abdelhedi
Université de Sfax, Faculté des Sciences de Sfax,
Département de Mathematiques, BP 1171, Sfax 3000, Tunisie.
Hatem Zaag
Université Sorbonne Paris Nord,
LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France

###### Abstract

We consider in this paper a perturbation of the standard semilinear heat
equation by a term involving the space derivative and a non-local term. In
some earlier works [1, 2], we constructed a solution $u$ for that equation
such that $u$ and $\nabla u$ both blow up at the origin and only there. We
also gave the final blow-up profile. In this paper, we refine our construction
method in order to get a sharper estimate on the gradient at blow-up.

AMS 2010 Classification: 35B20, 35B44, 35K55.
Keywords: Blow-up, nonlinear heat equation, gradient term, non-local term.

##  1 Introduction

We consider in this paper the following nonlinear parabolic equation

$\left\\{\begin{array}[]{lcl}u_{t}&=&\Delta u+|u|^{p-1}u+\mu|\nabla
u|\displaystyle\int_{B(0,|x|)}|u|^{q-1},\\\ u(0)&=&u_{0}\in
W^{1,\infty}(\mathbb{R}^{N}),\end{array}\right.$ (1.1)

where $u=u(x,t)\in\mathbb{R}$, $x\in\mathbb{R}^{N}$ and the parameters $p,\;q$
and $\mu$ are such that

$\displaystyle
p>3,\quad\frac{N}{2}(p-1)+1<q<\frac{N}{2}(p-1)+\frac{p+1}{2},\quad\mu\in\mathbb{R}.$
(1.2)

Equation $(\ref{eq_u})$ is wellposed in the weighted functional space
$W^{1,\infty}_{\beta}(\mathbb{R}^{N})$ defined as follows:

$W^{1,\infty}_{\beta}(\mathbb{R}^{N})=\\{g,\;(1+|y|^{\beta})g\in
L^{\infty},\;(1+|y|^{\beta})\nabla g\in L^{\infty}\\},$ (1.3)

where

$0\leq\beta<\frac{2}{p-1},\;\mbox{if}\;\mu=0\;\mbox{ and
}\;\displaystyle\frac{N}{q-1}<\beta<\frac{2}{p-1},\;\mbox{if}\;\mu\neq 0,$
(1.4)

as one may see from Appendix C in [1]. From the standard dichotomy, either the
maximal solution is global in time, or it exists up to some maximal time
$T<+\infty$ with

$\|u(t)\|_{W^{1,\infty}_{\beta}}\to\infty\mbox{ as }t\to+\infty.$

In that case, we say that $u(x,t)$ blows up in finite time, and we call $T$
the blow-up time of the solution.

When $\mu=0$, equation (1.1) becomes the standard semilinear heat equation
with power nonlinearity:

$u_{t}=\Delta u+|u|^{p-1}u.$ (1.5)

The existence of blow-up solutions for equation (1.5) has been extensively
studied, see Fujita [13], Ball [3], Berger and Kohn [4], Herrero and Velázquez
[21], Bricmont and Kupiainen [5], Merle and Zaag [25] and the references
therein.
In particular, the authors in [5] and [25] constructed a solution $u$ which
approaches an explicit universal profile $f$ depending only on $p$ and
independent from initial data as follows:

$\left\|(T-t)^{\frac{1}{p-1}}u(x,t)-f\left(\frac{x}{\sqrt{(T-t)|\log(T-t)|}}\right)\right\|_{L^{\infty}}\to
0,$ (1.6)

as $t\to T$, where $f$ is the profile defined by

$\displaystyle
f(z)=\left(p-1+\frac{(p-1)^{2}}{4p}|z|^{2}\right)^{-\frac{1}{p-1}}.$ (1.7)

The proof relies on 2 parts:
\- A formal part, where one finds an approximate solution, which will be
considered as a profile for the exact solution to be constructed.
\- A rigorous part, where one linearizes the equation around the approximate
solution (i.e. the profile), and shows that the linearized solution has a
solution which converges to $0$. This part relies itself on 2 steps: the
first, where we reduce the control of the solution (which is infinite
dimensional) to a finite dimensional problem. Then, the finite dimensional
problem is solved thanks to index theory.

This method has proved to be efficient for different PDEs from different
types, and no list can be exhaustive (see del Pino, Musso and Wei [7], Nouaili
and Zaag [32], Tayachi and Zaag [34], Duong and Zaag [8], Mahmoudi, Nouaili
and Zaag [23], Merle, Raphael, Rodnianski and Szeftel [24], Collot, Ghoul,
Nguyen and Masmoudi [6], etc…).

In [1] and [2], we considered equation (1.1) as a challenge for the
construction of blow-up solutions, since it presents a double difficulty: the
gradient term and the nonlocal term, and we were successful in proving the
following:

\- In [1], we constructed a solution $u(x,t)$ for equation $(\ref{eq_u})$
which blows up in finite time $T$ at $a=0$, and we proved that the solution
approaches the profile $f$ (1.7) in the sense that for all
$(x,t)\in\mathbb{R}^{N}\times[0,T)$:

$\displaystyle\displaystyle\Big{|}u(x,t)\\!-\\!(T\\!-\\!t)^{\\!-\\!\frac{1}{p-1}}f(\frac{x}{\sqrt{(T\\!-\\!t)|\log(T\\!-\\!t)|}})\Big{|}\\!\leq\\!\frac{C}{1\\!+\\!(\frac{|x|^{2}}{T\\!-t})^{\frac{\beta}{2}}}\frac{(T\\!-\\!t)^{\\!-\\!\frac{1}{p-1}}}{|\log(T\\!-\\!t)|^{\frac{1\\!-\\!\beta}{2}}},$
(1.8)

and

$\displaystyle\displaystyle\\!\Big{|}\\!\nabla\\!u(x,t)\\!-\\!\frac{(T\\!-\\!t)^{\\!-\\!\frac{1}{2}\\!-\\!\frac{1}{p\\!-\\!1}}}{\sqrt{|\log(T\\!-\\!t)|}}\nabla\\!f\\!(\\!\frac{x}{\sqrt{(T\\!\\!-\\!t)|\log(T\\!\\!-\\!t)|}})\Big{|}\\!\\!\leq\\!\\!\\!\frac{C}{1\\!\\!+\\!\\!(\frac{|x|^{2}}{T\\!-t})^{\\!\frac{\beta}{2}}}\frac{(T\\!-\\!t)^{\\!-\\!\frac{1}{2}\\!-\\!\frac{1}{p-1}}}{|\log(\\!T-\\!t)|^{\frac{1\\!-\\!\beta}{2}}},$
(1.9)

(note that $0\leq\beta<1$ from (1.4) and (1.2)), and that $f$ is called the
“intermediate” profile).

\- In [2], adapting the technique developed for equation (1.5) by Giga and
Kohn [16], we prove that neither $u$ nor $\nabla u$ blow up outside the
origin. Since $u$ blows up at the origin by (1.8), this implies the single
point blow-up property for $u$. Using the mean value theorem, this yields that
$\nabla u$ blows up at the origin, which proves the single point blow-up
property for $\nabla u$ too. We also prove the existence of a blow-up final
profile $u^{\ast}$ such that $u(x,t)\to u^{\ast}(x)$ as $t\to T$ in $C^{1}$ of
every compact of $\mathbb{R}^{N}\backslash\\{0\\}$. Next, we find an
equivalent of $u^{\ast}$ and an upper bound on $\nabla u^{*}$ near the blow-up
point:

$u^{\ast}(x)\sim\Big{[}\frac{8p|\log|x||}{(p-1)^{2}|x|^{2}}\Big{]}^{\frac{1}{p-1}},\;\mbox{as}\;x\to
0,$ (1.10)

and for $|x|$ small,

$\displaystyle|\nabla u^{\ast}(x)|\leq
C|x|^{-\frac{p+1}{p-1}}\big{|}\log|x|\big{|}^{\frac{p+3}{4(p-1)}}.$ (1.11)

Our goal in this paper is to give a refined asymptotic description of the
blow-up solution. In fact, we prove more refined versions of (1.8), (1.9) and
(1.11).
This will be done through the introduction of a sharper shrinking set. More
precisely, we prove the following theorem:

######  THEOREM 1.1 (A sharper description of blow-up).

Let $\mu\in\mathbb{R}$, $p>3,$ and $q\in\mathbb{R}$ such that
$\displaystyle\frac{N}{2}(p-1)+1<q<\frac{N}{2}(p-1)+\frac{p+1}{2}$.
Consider an arbitrary $\beta$ such that

$0\leq\beta<\frac{2}{p-1},\;\mbox{if}\;\mu=0\;\mbox{ and
}\;\displaystyle\frac{N}{q-1}<\beta<\frac{2}{p-1},\;\mbox{if}\;\mu\neq 0.$
(1.12)

Consider also some arbitrary $\varepsilon_{1}\in(0,\frac{1}{2}]$ and
$\alpha\in(0,\frac{1}{2})$.
Then, there exists $T>0$ such that equation $(\ref{eq_u})$ has a solution
$u(x,t)$ such that $u$ and $\nabla u$ simultaneously blow up at time $T$ at
the point $a=0$. Moreover,

  1. 1.

For all $t\in[0,T)$, for all $x\in\mathbb{R}^{N}$,

$\displaystyle|u(x,t)-(T-t)^{-\frac{1}{p-1}}f(\frac{x}{\sqrt{(T-t)|\log(T-t)|}})|\leq\frac{C}{1+(\frac{|x|^{2}}{T-t})^{\frac{\beta}{2}}}\frac{(T-t)^{-\frac{1}{p-1}}}{|\log(T-t)|^{1-\frac{\beta}{2}-\varepsilon_{1}}},$
(1.13)

and

$\displaystyle\\!\\!|\nabla
u(x,t)-\frac{(T\\!-\\!t)^{\\!-\frac{1}{2}\\!-\\!\frac{1}{p\\!-\\!1}}}{\sqrt{|\log(T-t)|}}\nabla
f(\frac{x}{\sqrt{(T\\!-\\!t)|\log(T\\!-\\!t)|}})|\\!\leq\\!\\!\frac{C}{1\\!+\\!(\frac{|x|^{2}}{T-t})^{\frac{\beta}{2}}}\frac{(T\\!-\\!t)^{-\frac{1}{2}-\frac{1}{p-1}}}{|\log(T\\!-\\!t)|^{1\\!-\\!\frac{\beta}{2}\\!-\\!\varepsilon_{1}}},$
(1.14)

where $f$ is defined in (1.7).

  2. 2.

Blow-up occurs only at the origin.

  3. 3.

For all $x\neq 0$, $u(x,t)\to u^{\ast}(x)$ as $t\to T$ in $C^{1}$ of every
compact of $\mathbb{R}^{N}\setminus\\{0\\}$, with

$u^{\ast}(x)\sim\Big{[}\frac{8p|\log|x||}{(p-1)^{2}|x|^{2}}\Big{]}^{\frac{1}{p-1}},\;\mbox{as}\;x\to
0,$ (1.15)

and for $|x|$ small,

$\displaystyle|\nabla u^{\ast}(x)|\leq
C|x|^{-\frac{p+1}{p-1}}\big{|}\log|x|\big{|}^{\frac{p+1}{2(p-1)}-\alpha}.$
(1.16)

######  REMARK 1.2.

  1. 1.

With respect to our earlier work [1] and [2], the improvement lays in (1.13),
(1.14) and  (1.16), since we could obtain the result in those papers only with
$\varepsilon_{1}=\frac{1}{2}$ and $\alpha=\frac{1}{4}$, whereas we are able
now to get it for smaller values of $\varepsilon_{1}$ and larger values of
$\alpha$, namely for $\varepsilon_{1}$ arbitrarily close to $0$ and $\alpha$
arbitrarily close to $\frac{1}{2}$.

  2. 2.

As we have already noted in our earlier paper [1], already when $\mu=0$, our
error estimate in (1.13) is better than in [25], thanks to the term
$1+(\frac{|x|^{2}}{T-t})^{\frac{\beta}{2}}$ in the denominator. In fact, our
estimate is even better than (1.8) proved in [1], since we can take
$\varepsilon_{1}$ arbitrarily small and not just equal to $\frac{1}{2}$ as in
(1.8).

  3. 3.

As we have already noted in our previous paper [2], the blow-up of the
gradient at the origin is a consequence of the single-point blow-up of the
solution and the mean value theorem. In this paper, we do better, by
exhibiting some $\xi(t)\to 0$ as $t\to T$, so that
$|\nabla(\xi(t),t)|\to\infty$ (see below in page 4). The improvement in (1.14)
is crucial in obtaining that result, keeping in mind that the version with
$\varepsilon_{1}=\frac{1}{2}$ given in [1] (see (1.9) above) was not enough to
derive such a sequence $\xi(t)$.

  4. 4.

We wonder whether one may be able to get an equivalent for the gradient final
blow-up profile. Such a question seems to be hard, since, up to our knowledge,
no such equivalent was proved in the literature, except in the recent work on
the standard semilinear heat equation (1.5) by Duong, Ghoul and Zaag [9].

The paper is organized as follows. In Section $2$, we give a formulation of
the problem. In Section $3$, we prove the existence of a solution of equation
$(\ref{eq_v})$. Finally, in Section $4$, we prove Theorem 1.1.

##  2 Formulation of the problem

In this section, we consider $T>0$ and $\varepsilon_{1}\in(0,\frac{1}{2}]$,
then we recall the following similarity variables transformation introduced by
Giga and Kohn [16], [17], [18]:

$\displaystyle
y=\frac{x}{\sqrt{T-t}},\;\;s=-\log(T-t)\;\;\mbox{and}\;\;w(y,s)=\displaystyle(T-t)^{\frac{1}{p-1}}u(x,t),$
(2.1)

where $T$ is the time where we want to make the solution blow up.
If $u(x,t)$ satisfies $(\ref{eq_u})$ for all
$(x,t)\in\mathbb{R}^{N}\times[0,T)$, then $w(y,s)$ satisfies the following
equation for all $(y,s)\in\mathbb{R}^{N}\times[-\log T,+\infty)$:

$\partial_{s}w=\Delta w-\frac{1}{2}y\cdot\nabla
w-\frac{1}{p-1}w+|w|^{p-1}w+\mu e^{-\gamma s}|\nabla
w|\int_{B(0,|y|)}|w|^{q-1},$ (2.2)

where $\gamma=\displaystyle\frac{p-q}{p-1}+\frac{N-1}{2}$.
Note from (1.2) that $\gamma>0$, which explains the little effect of the
perturbation term for large time $s$. This allows us to adopt a perturbative
approach with respect to earlier constructions given for the standard
semilinear heat equation (1.5).

With the transformation (2.1), we reduce the construction of a solution
$u(x,t)$ for equation (1.1) that blows up at $T<\infty$ to the construction of
a global solution $w(y,s)$ for equation (2.2) such that

$\forall s\geq
s_{0},\;\;\varepsilon_{0}\leq\|w(s)\|_{L^{\infty}}\leq\frac{1}{\varepsilon_{0}},$

for some $\varepsilon_{0}>0$. In fact, we will require more information on the
solution, in the sense that we would like to find $s_{0}>0$ and initial data
$w_{0}$ such that the solution $w$ of equation (2.2) with $w(s_{0})=w_{0}$
satisfies

$\|w(y,s)-f(\frac{y}{\sqrt{s}})\|_{W^{1,\infty}_{\beta}}\to_{s\to\infty}0,$

where $f$ is the profile defined by

$\displaystyle f(z)=(p-1+\frac{(p-1)^{2}}{4p}|z|^{2})^{-\frac{1}{p-1}}.$ (2.3)

As we have already pointed-out in [1], we don’t linearize (2.2) around
$f(\frac{y}{\sqrt{s}})+\frac{\kappa N}{2ps}$ as in the case of equation (1.5)
treated in [25], since this function is not in the space
$W^{1,\infty}_{\beta}$. In fact, we linearize equation $(\ref{eq_w})$ around a
new profile

$\displaystyle\varphi(y,s)=f(\frac{y}{\sqrt{s}})+\frac{\kappa
N}{2ps}\chi_{0}(\frac{y}{g_{\varepsilon}(s)}),$ (2.4)

where $\kappa=(p-1)^{-\frac{1}{p-1}}$ is a stationary solution for equation
$(\ref{eq_w})$, $\chi_{0}\in C_{0}^{\infty}$ with ${\rm supp}(\chi_{0})\subset
B(0,2)$, $\chi_{0}\equiv 1$ on $B(0,1)$ and
$g_{\varepsilon}(s)=s^{\frac{1}{2}+\varepsilon}$, where $\varepsilon$ is a
fixed constant satisfying

$0<\varepsilon<\min(1,\varepsilon_{1}\beta^{-1}),$ (2.5)

(we could have taken
$\varepsilon=\frac{1}{2}\min(1,\varepsilon_{1}\beta^{-1})$).
Introducing

$v(y,s)=w(y,s)-\varphi(y,s),$ (2.6)

the problem is reduced to constructing a function $v$ such that

$\lim_{s\to+\infty}\|v(s)\|_{W^{1,\infty}_{\beta}}=0.$

If $w$ satisfies equation $(\ref{eq_w})$, then $v$ satisfies the following
equation

$\partial_{s}v=(\mathcal{L}+V)v+B(v)+R(y,s)+\mathcal{N}(y,s),$ (2.7)

where

  * •

the linear term $(\mathcal{L}+V)v$ is given by

$\mathcal{L}(v)=\Delta v-\frac{1}{2}y\cdot\nabla v+v\;\mbox{ with
}\;V(y,s)=p\varphi^{p-1}-\frac{p}{p-1},$ (2.8)

  * •

the nonlinear term is

$B(v)=|v+\varphi|^{p-1}(v+\varphi)-\varphi^{p}-p\varphi^{p-1}v,$ (2.9)

  * •

the remainder term involving $\varphi$ is

$R(y,s)=\Delta\varphi-\frac{1}{2}y\cdot\nabla\varphi-\frac{1}{p-1}\varphi+\varphi^{p}-\partial_{s}\varphi,$
(2.10)

  * •

and the “new” term is

$\displaystyle\mathcal{N}(y,s)=\mu e^{-\gamma s}|\nabla
v+\nabla\varphi|\int_{B(0,|y|)}|v+\varphi|^{q-1}.$ (2.11)

Let us introduce the following integral form of equation $(\ref{eq_v})$, for
each $s\geq\sigma\geq s_{0}$:

$v(s)=K(s,\sigma)v(\sigma)+\int_{\sigma}^{s}K(s,t)\big{(}B(v(t))+R(t)+\mathcal{N}(t)\big{)}dt,$
(2.12)

where $K$ is the fundamental solution of the operator $\mathcal{L}+V$.
Since the dynamics of (2.7) are influenced by the linear operator
$\mathcal{L}+V$, we first need to recall some of its properties (for more
details, see [5]).
The operator $\mathcal{L}$ is self-adjoint in $D(\mathcal{L})\subset
L^{2}_{\rho}(\mathbb{R}^{N})$, where

$L^{2}_{\rho}(\mathbb{R}^{N})=\\{v\in
L^{2}_{loc}(\mathbb{R}^{N});\quad\int_{\mathbb{R}^{N}}(v(y))^{2}\rho(y)dy<\infty\\},\quad\rho(y)=\displaystyle\frac{e^{-\frac{|y|^{2}}{4}}}{(4\pi)^{\frac{N}{2}}}.$

The spectrum of $\mathcal{L}$ consists only in eigenvalues given by

${\rm spec}(\mathcal{L})=\\{1-\frac{m}{2};\quad m\in\mathbb{N}\\},$

and its eigenfunctions are rescaled Hermite polynomials.
If $N=1$, all the eigenvalues are simple, and the eigenfunction corresponding
to $1-\frac{m}{2}$ is

$\displaystyle
h_{m}(y)=\sum_{k=0}^{[\frac{m}{2}]}\frac{m!}{k!(m-2k)!}(-1)^{k}y^{m-2k}.$
(2.13)

In particular, $h_{0}(y)=1$, $h_{1}(y)=y$ and $h_{2}(y)=y^{2}-2$ are the
eigenfunctions corresponding to nonnegative eigenvalues
$\lambda\in\\{1,\frac{1}{2},0\\}$. We remark that $h_{m}$ satisfies:

$\displaystyle\int_{\mathbb{R}}h_{n}h_{m}\rho dx=2^{n}n!\delta_{n,m}.$

We also introduce $\displaystyle
k_{m}=\frac{h_{m}}{\|h_{m}\|_{L^{2}_{\rho}(\mathbb{R})}^{2}}$.
If $N\geq 2$, the eigenspace corresponding to $1-\frac{m}{2}$ is given by

$E_{m}=\\{h_{m_{1}}(y_{1})\cdots h_{m_{N}}(y_{N});\quad\quad m_{1}+\cdots
m_{N}=m\\}.$

In particular, these are the eigenspaces corresponding to nonnegative
eigenvalues:

$E_{0}=\\{1\\},\;E_{1}=\\{y_{i};\quad\;i=1,\cdots,N\\}\;\mbox{
and}\;E_{2}=\\{h_{2}(y_{i}),\;y_{i}y_{j};\quad i,j=1,\cdots,N,\;i\neq j\\}.$

The potential $V(y,s)$ has two fundamental properties:

  * •

$V(.,s)\to 0$ in $L^{2}_{\rho}$ as $s\to+\infty$. In particular, on bounded
sets or in the “blow-up area” $\\{|y|\leq K_{0}\sqrt{s}\\}$, where $K_{0}>0$
will be fixed large enough, the effect of $V$ is considered as a perturbation
of the effect of $\mathcal{L}$, except maybe on the null eigenspace of
$\mathcal{L}$.

  * •

Outside the “blow-up area”, we have the following property: for all $\eta>0$
the exist $C_{\eta}>0$ and $s_{\eta}$ such that

$\displaystyle\sup_{s\geq s_{\eta},|y|\geq
C_{\varepsilon}\sqrt{s}}|V(y,s)-(-\frac{p}{p-1})|\leq\eta.$

This means that $\mathcal{L}+V$ behaves like $\mathcal{L}-\frac{p}{p-1}$ in
the region $\\{|y|\geq K_{0}\sqrt{s}\\}$ for $K_{0}$ large enough. Since
$-\frac{p}{p-1}<-1$ and $1$ is the largest eigenvalue of $\mathcal{L}$, we can
consider the operator $\mathcal{L}+V$ outside the blow-up area as an operator
with a purely negative spectrum, which greatly simplifies the analysis.
Bearing in mind that the behavior of $V$ is not the same inside and outside
the “blow-up” area, we decompose $v$ as follows. Let us first introduce the
following cut-off function:

$\chi(y,s)=\chi_{0}(\displaystyle\frac{|y|}{K_{0}\sqrt{s}}),$ (2.14)

where $K_{0}>0$ will be fixed large enough so that various technical estimates
hold, and the cut-off function $\chi_{0}$ was already introduced after (2.4).
We write

$v(y,s)=v_{b}(y,s)+v_{e}(y,s),$ (2.15)

with

$v_{b}(y,s)=v(y,s)\chi(y,s),\quad v_{e}(y,s)=v(y,s)(1-\chi(y,s)).$ (2.16)

We note that ${\rm supp}\;v_{b}(s)\subset B(0,2K_{0}\sqrt{s})$ and ${\rm
supp}\;v_{e}(s)\subset\mathbb{R}^{N}\backslash B(0,K_{0}\sqrt{s})$.
In order to control $v_{b}$, we decompose it according to the sign of the
eigenvalues of $\mathcal{L}$ as follows (for simplicity, we give the
decomposition only for $N=1$, bearing in mind that the situation for $N\geq 2$
is the same, except for some more complicated notations that can be found in
Nguyen and Zaag [29], where the formalism in higher dimensions is extensively
given):

$\displaystyle
v(y,s)=v_{b}(y,s)+v_{e}(y,s)=\sum_{m=0}^{2}v_{m}(s)h_{m}(y)+v_{-}(y,s)+v_{e}(y,s),$
(2.17)

where for $0\leq m\leq 2$, $v_{m}=P_{m}(v_{b})$ and $v_{-}(s)=P_{-}(v_{b})$,
with $P_{m}$ being the $L^{2}_{\rho}$ projector on $h_{m}$, the eigenfunction
corresponding to $\lambda=1-\frac{m}{2}$, and $P_{-}$ the projector on
$\\{h_{i}\;|\;i\geq 3\\}$, the nonpositive subspace of the operator
$\mathcal{L}$.

##  3 Existence

This section is devoted to the proof of the existence of a solution $v$ of
$(\ref{eq_v})$ such that

$\displaystyle\lim_{s\to+\infty}\|v(s)\|_{W^{1,\infty}_{\beta}}=0.$ (3.1)

We proceed exactly as in our previous paper [1], except for the definition of
the shrinking set below in Definition 3.1, where we will introduce a smaller
set, leading to a sharper estimate on the norm in (3.1), at the expense of
more involved estimates. For simplicity in the notations, we give the proof
only when $N=1$.

As in our previous paper [1], our argument is a non trivial adaptation of the
method performed of equation (1.5) by Bricmont and Kupiainen [5], Merle and
Zaag [25], Nguyen and Zaag [31] and Tayachi and Zaag [34]. Accordingly, as in
[1], we proceed in 4 steps:

  * •

In the first step, we define a new shrinking set
$\mathcal{V}_{A,\varepsilon_{1}}(s)$ and translate our goal in (3.1) to the
construction of a solution belonging to that set.

  * •

In the second step, using the spectral properties, we reduce our goal from the
control of $v(s)$ (an infinite dimensional variable) in
$\mathcal{V}_{A,\varepsilon_{1}}(s)$ to the control of its two first
components $(v_{0}(s),v_{1}(s))$ in $[-\frac{A}{s^{2}},\frac{A}{s^{2}}]^{2}$
(a two-dimensional variable).

  * •

In the third step, we prove a parabolic regularity estimate, in order to
obtain the convergence of the gradient term.

  * •

In the last step, we solve the two-dimensional problem using index theory and
we prove the existence of a solution $v$ of (2.7) belonging to the new
shrinking set $\mathcal{V}_{A,\varepsilon_{1}}(s)$.

In the following, we fix $\beta$ satisfying (1.4) and we denote by $C$ a
generic positive constant, depending only on $p$, $\mu$, $\beta$ and $K_{0}$.
Note that $C$ does not depend on $A$, $\varepsilon_{1}$ and $s_{0}$, the
constants that will appear below.

###  3.1 A New shrinking set $\mathcal{V}_{\beta,K_{0},A,\varepsilon_{1}}(s)$

In this subsection, we introduce the new shrinking set, which is the main
novelty with respect to our previous work [1]:

######  DEFINITION 3.1 (A set shriking to zero).

For all $K_{0}\geq 1$, $A\geq 1$, $0<\varepsilon_{1}\leq\frac{1}{2}$ and
$s\geq 1$, we define $\mathcal{V}_{\beta,K_{0},A,\varepsilon_{1}}(s)$ (or
$\mathcal{V}_{A,\varepsilon_{1}}(s)$ for simplicity) as the set of all
functions $g$ such that $(1+|y|^{\beta})g\in L^{\infty}(\mathbb{R}^{N})$ and

$\displaystyle|g_{k}|\leq\displaystyle\frac{A}{s^{2}},\;k=0,1,\;\;|g_{2}|\leq\displaystyle\frac{A^{2}\log
s}{s^{2}},\;\;\|\displaystyle\frac{g_{-}}{1+|y|^{3}}\|_{L^{\infty}}\leq\displaystyle\frac{A}{s^{\frac{5}{2}-\varepsilon_{1}}},$
(3.2) $\displaystyle\|\displaystyle
g_{e}\|_{L^{\infty}}\leq\displaystyle\frac{A^{2}}{s^{1-\varepsilon_{1}}},\;\;\|\displaystyle(1+|y|^{\beta})g_{e}\|_{L^{\infty}}\leq\displaystyle\frac{A^{2}}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}},$
(3.3)

where $g_{k}$, $g_{-}$ and $g_{e}$ are defined in (2.16) and (2.17).

######  REMARK 3.2.

  1. 1.

Since $p>3$ and $\beta$ satisfy (1.4), it follows that
$\beta<\frac{2}{p-1}<1$. Therefore, since
$\varepsilon_{1}\in(0,1-\frac{\beta}{2})$, it follows that the bounds on
$g_{-}$ and $g_{e}$ go to zero as $s\to\infty$.

  2. 2.

When $\varepsilon_{1}=\frac{1}{2}$, we recover the shrinking set of our
previous paper [1]. Since we will take $\varepsilon_{1}$ arbitrarily small
here, we clearly improve our estimates. This is the key novelty of our work.

Since this shrinking set is different from all the previous studies, we need
to introduce new estimates. Let us first note that the set
$\mathcal{V}_{A,\varepsilon_{1}}(s)$ is increasing (for fixed
$s,\;\beta,\;K_{0},\;\varepsilon_{1}$) with respect to $A$ in the sense of
inclusion. Let us now give some properties of the shrinking set in the
following:

######  PROPOSITION 3.3.

For all $K_{0}\geq 1$, $A\geq 1$ and $0<\varepsilon_{1}\leq\frac{1}{2}$, there
exists $s_{1}(K_{0},A,\varepsilon_{1})$ such that, for all $s\geq
s_{1}(K_{0},A,\varepsilon_{1})$ and $g\in\mathcal{V}_{A,\varepsilon_{1}}(s)$,
we have

  * i)

$\|g\|_{L^{\infty}(|y|\leq
2K_{0}\sqrt{s})}\leq\displaystyle\frac{CA}{s^{1-\varepsilon_{1}}}\quad\mbox{and}\quad\|g\|_{L^{\infty}(\mathbb{R})}\leq\displaystyle\frac{CA^{2}}{s^{1-\varepsilon_{1}}}.$
(3.4)

  * ii)

$\|(1+|y|^{\beta})g\|_{L^{\infty}(|y|\leq
2K_{0}\sqrt{s})}\leq\displaystyle\frac{CA}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}\;\;\mbox{and}\;\;\|(1+|y|^{\beta})g\|_{L^{\infty}(\mathbb{R})}\leq\displaystyle\frac{CA^{2}}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$
(3.5)

######  REMARK 3.4.

Here and throughout the paper, the generic constant $C$ depends on $p$ and
$N$, and may depend on $K_{0}$ too.

We omit the proof of this proposition, since it is obtained by a simple
modification of the proof of Proposition $3.8$ in [1] page $14$.

The construction of a solution $v$ of (2.7) is based on a careful choice of
initial data at time $s_{0}=-\log T$, as we do in the following:

######  DEFINITION 3.5.

(Choice of the initial data) For all $K_{0}\geq 1$, $A\geq 1$, $s_{0}=-\log
T>1$ and $d_{0},\,d_{1}\in\mathbb{R}$, we consider the following function as
initial data for equation $(\ref{eq_v})$:

$\psi_{s_{0},d_{0},d_{1}}(y)=\displaystyle\frac{A}{s_{0}^{2}}(d_{0}h_{0}(y)+d_{1}h_{1}(y))\chi(2y,s_{0}),$
(3.6)

where $h_{i}$, $i=0,1$ are defined in $(\ref{vect_prop})$ and $\chi$ is
defined in $(\ref{chi})$.

Reasonably, we choose the parameter $(d_{0},d_{1})$ such that the initial data
$\psi_{s_{0},d_{0},d_{1}}\in\mathcal{V}_{A,\varepsilon_{1}}(s_{0})$. More
precisely, we claim the following result:

######  PROPOSITION 3.6.

(Properties of initial data) For each $0<\varepsilon_{1}\leq\frac{1}{2}$,
$K_{0}\geq 1$, $A\geq 1$, there exists $s_{01}(K_{0},A,\varepsilon_{1})>1$
such that for all $s_{0}\geq s_{01}(K_{0},A,\varepsilon_{1})$:

  * i)

There exists a rectangle $\mathcal{D}_{s_{0}}\subset[-2,2]^{2}$ such that the
mapping

$\begin{array}[]{lcl}\Phi:\mathbb{R}^{2}&\to&\mathbb{R}^{2}\\\
(d_{0},d_{1})&\mapsto&(\psi_{0},\psi_{1}),\end{array}$ (3.7)

(where $\psi:=\psi_{s_{0},d_{0},d_{1}}$) is linear, one to one from
$\mathcal{D}_{s_{0}}$ onto
$[\displaystyle-\frac{A}{s_{0}^{2}},\displaystyle\frac{A}{s_{0}^{2}}]^{2}$ and
maps $\partial\mathcal{D}_{s_{0}}$ into
$\partial([\displaystyle-\frac{A}{s_{0}^{2}},\displaystyle\frac{A}{s_{0}^{2}}]^{2})$.
Moreover, it has degree one on the boundary.

  * ii)

For all $(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$,
$\psi\in\mathcal{V}_{A,\varepsilon_{1}}(s_{0})$ with strict inequalities
except for $(\psi_{0},\psi_{1})$, in the sense that

$\psi_{e}\equiv
0,\quad|\psi_{-}(y)|<\frac{1}{s_{0}^{\frac{5}{2}-\varepsilon_{1}}}(1+|y|^{3}),\;\forall
y\in\mathbb{R},\\\
|\psi_{k}|\leq\frac{A}{s_{0}^{2}},\;k=0,1,\quad|\psi_{2}|<\displaystyle\frac{\log
s_{0}}{s_{0}^{2}}.$ (3.8)

  * iii)

Moreover, for all $(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$, we have

$\displaystyle\|(1+|y|^{\beta})\nabla\psi\|_{L^{\infty}}$ $\displaystyle\leq$
$\displaystyle\frac{CA}{s_{0}^{2-\frac{\beta}{2}}}\leq\frac{1}{s_{0}^{1-\frac{\beta}{2}-\varepsilon_{1}}},$
(3.9) $\displaystyle|\nabla\psi_{-}(y)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{s_{0}^{\frac{5}{2}-\varepsilon_{1}}}(1+|y|^{3}).$
(3.10)

We omit the proof, since it is obtained by a simple modification of the proof
of Proposition $4.5$ of [34]. For more details, we refer the interested reader
to pages $5915-5918$ of [34], and pages $14-15$ of [1].

###  3.2 Parabolic Regularity

As stated above, our goal is to get the convergence in $W^{1,\infty}_{\beta}$.
Thus, we need to prove that the solution $v$ of (2.7) satisfies

$\displaystyle\lim_{s\to+\infty}\|(1+|y|^{\beta})\nabla
v(s)\|_{L^{\infty}}=0.$

More precisely, we need the following parabolic regularity result for equation
$(\ref{eq_v})$:

######  PROPOSITION 3.7.

(Parabolic regularity in $\mathcal{V}_{A,\varepsilon_{1}}(s)$)
For all $0<\varepsilon_{1}\leq\frac{1}{2}$, $K_{0}\geq 1$ and $A\geq 1$, there
exists $s_{02}(K_{0},A,\varepsilon_{1})$ such that for all $s_{0}\geq
s_{02}(K_{0},A)$, if $v$ is the solution of $(\ref{eq_v})$ for all
$s\in[s_{0},s_{1}]$, $s_{0}\leq s_{1}$, with initial data at $s_{0}$, given in
$(\ref{ci})$ with $(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$ defined in Proposition
3.6, and $v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$, then, for all
$s\in[s_{0},s_{1}]$, we have

$\displaystyle\|\displaystyle\nabla
v(s)\|_{L^{\infty}}\leq\displaystyle\frac{CA^{2}}{s^{1-\varepsilon_{1}}}\quad\mbox{and}\quad\|\displaystyle(1+|y|^{\beta})\nabla
v(s)\|_{L^{\infty}}\leq\displaystyle\frac{CA^{2}}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$
(3.11)

The proof of the previous proposition follows exactly as in [1]. For that
reason, the proof is omitted and we refer the reader to subsection 3.2.3 page
21-24 in [1] for details.

Since $v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$, we clearly see from
$(\ref{est_shrin})$ and $(\ref{reg})$ that

$\|\displaystyle(1+|y|^{\beta})v(s)\|_{L^{\infty}}+\|\displaystyle(1+|y|^{\beta})\nabla
v(s)\|_{L^{\infty}}\leq\displaystyle\frac{CA^{2}}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$
(3.12)

###  3.3 Reduction to a finite dimensional problem

The following section is crucial in the proof of the existence of the blow-up
solution. In fact, we reduce here the construction question to a finite
dimensional problem. As in [27], [10], [34] and [1], we prove that it is
enough to control the $2$ components
$(v_{0}(s),v_{1}(s))\in[\displaystyle-\frac{A}{s^{2}},\displaystyle\frac{A}{s^{2}}]^{2}$
in order to control the solution $v(s)$ in
$\mathcal{V}_{A,\varepsilon_{1}}(s)$, which is infinite dimensional.
Precisely, we have the following proposition.

######  PROPOSITION 3.8.

(Control of $v(s)$ by $(v_{0},v_{1})(s)$ in
$\mathcal{V}_{A,\varepsilon_{1}}(s)$ ) For all
$0<\varepsilon_{1}\leq\frac{1}{2}$, there exists $K_{3}(\varepsilon_{1})\geq
1$ such that for any $K_{0}\geq K_{3}(\varepsilon_{1})$, there exists
$A_{3}(K_{0},\varepsilon_{1})\geq 1$ such that for each $A\geq
A_{3}(K_{0},\varepsilon_{1})$, there exists
$s_{03}(K_{0},A,\varepsilon_{1})\in\mathbb{R}$ such that for all $s_{0}\geq
s_{03}(K_{0},A,\varepsilon_{1})$, the following holds:
If $v$ is a solution of $(\ref{eq_v})$ with initial data at $s=s_{0}$ given by
$(\ref{ci})$ with $(d_{0},d_{1})\in\mathcal{D}_{s_{0}}$, and
$v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$ for all $s\in[s_{0},s_{1}]$, with
$v(s_{1})\in\partial\mathcal{V}_{A,\varepsilon_{1}}(s_{1})$ for some
$s_{1}\geq s_{0}$, then:

  * i)

(Reduction to a finite dimensional problem) We have:

$(v_{0}(s_{1}),v_{1}(s_{1}))\in\partial([\displaystyle-\frac{A}{s_{1}^{2}},\displaystyle\frac{A}{s_{1}^{2}}]^{2}).$

  * ii)

(Transverse crossing) There exist $m\in\\{0,1\\}$ and $\omega\in\\{-1,1\\}$
such that

$\omega
v_{m}(s_{1})=\displaystyle\frac{A}{s_{1}^{2}}\quad\mbox{and}\quad\omega
v^{\prime}_{m}(s_{1})>0.$

The remainder of this section is devoted to the proof of Proposition 3.8. We
proceed in $2$ steps:

  * •

In the first step, we prove that if
$v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$, then $B(v)$, $R(y,s)$ and
$\mathcal{N}(y,s)$ given in $(\ref{eq_v})$ belong
$\mathcal{V}_{C,\varepsilon_{1}}(s)$ and the potential term
$Vv(s)\in\mathcal{V}_{CA,\varepsilon_{1}}(s)$, for some positive constant $C$.

  * •

In the second step, we prove the result of the reduction to a finite
dimensional problem.

####  3.3.1 Step 1: Preliminary estimates on various terms of equation
$(\ref{eq_v})$

In the following, we prove that the remainder term $R(y,s)$ is in
$\mathcal{V}_{C,\varepsilon_{1}}(s)$ for $s$ large enough and some $C>0$. We
also prove that if $v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$, then the
nonlinear term $B(v)\in\mathcal{V}_{C,\varepsilon_{1}}(s)$ and the potential
term $Vv$ is trapped in $\mathcal{V}_{CA,\varepsilon_{1}}(s)$. Furthermore,
under some additional assumptions on $v$, we prove that the new term
$\mathcal{N}(y,s)$ is also trapped in $\mathcal{V}_{C,\varepsilon_{1}}(s)$.
More precisely, we prove the following result:

######  LEMMA 3.9.

  1. 1.

For all $K_{0}\geq 1$ and $0<\varepsilon_{1}\leq\frac{1}{2}$, there exists
$s_{3}(K_{0},\varepsilon_{1})$ sufficiently large such that for all $s\geq
s_{3}(K_{0},\varepsilon_{1})$, the remainder term
$R(s)\in\mathcal{V}_{C,\varepsilon_{1}}(s)$.

  2. 2.

For all $K_{0}\geq 1$, $0<\varepsilon_{1}\leq\frac{1}{2}$ and $A\geq 1$, there
exists $s_{4}(K_{0},A,\varepsilon_{1})$ sufficiently large such that for all
$s\geq s_{4}(K_{0},A,\varepsilon_{1})$, if
$v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$, then the nonlinear term
$B(v)\in\mathcal{V}_{C,\varepsilon_{1}}(s)$ and the potential term
$Vv\in\mathcal{V}_{CA,\varepsilon_{1}}(s)$.

We need the following technical lemma before proving Lemma 3.9.

######  LEMMA 3.10.

For all $K_{0}\geq 1$, there exists $s_{2}(K_{0})$ such that for all $s\geq
s_{2}(K_{0})$, the following holds:
(i) if $g(y)=1$ then
$\left\|\displaystyle\frac{g_{-}(y)}{1+|y|^{3}}\right\|_{L^{\infty}}\leq\frac{C}{K_{0}^{3}s^{\frac{3}{2}}}$,
(ii) if $g(y)=y^{2}$, then
$\left\|\displaystyle\frac{g_{-}(y)}{1+|y|^{3}}\right\|_{L^{\infty}}\leq\frac{C}{K_{0}\sqrt{s}}$.

###### Proof.

The proof of this technical lemma follows exactly as in [34]. For that reason,
the proof is omitted, and we refer the interested reader to Lemma $4.8$ page
$5916-5917$ in [34] for a similar case. ∎

Now, we are ready to prove Lemma 3.9.

###### Proof of Lemma 3.9.

In comparison with [1], the estimates for $B(v)$ and $vV$ can be adapted
smoothly (see subsection 3.2.2 page $14-17$ in [1] and subsection 4.2.2 page
$5918-5923$ in [34]), unlike for $R(y,s)$. For that reason, we only prove the
estimate for the latter. Moreover, since the constraints on $R_{m}(s)$ for
$m=0,1,2$ in our new shrinking set are the same as with the old shrinking set
in [1] (see Remark 3.2), we refer the reader to Lemma 3.9 page 15 in that
paper for the proof of the estimates on those components, and focus only on
the proofs of the estimates related to $R_{-}(s)$ and $R_{e}(s)$.
Since $-\frac{1}{2}zf^{\prime}(z)-\frac{1}{p-1}f(z)+f(z)^{p}=0$ by definition
(2.3) of $f(z)$, we write $R(y,s)$ introduced in (2.10) as follows

$\displaystyle R(y,s)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{s}f^{\prime\prime}(z)+\frac{1}{2s}zf^{\prime}(z)+\big{(}f(z)+\chi_{0}(Z)\frac{\kappa}{2ps}\big{)}^{p}-f(z)^{p}$
(3.13) $\displaystyle+$
$\displaystyle\frac{\kappa}{2ps}\Big{[}\frac{1}{g_{\varepsilon}^{2}}\chi_{0}^{\prime\prime}(Z)-\big{(}\frac{1}{2}-\frac{g_{\varepsilon}^{\prime}(s)}{g_{\varepsilon}(s)}\big{)}Z\chi_{0}^{\prime}(Z)+(\frac{1}{s}-\frac{1}{p-1})\chi_{0}(Z)\Big{]}$
$\displaystyle=$ $\displaystyle R_{i}+R_{ii},$

where $\displaystyle z=\frac{y}{\sqrt{s}},\;Z=\frac{y}{g_{\varepsilon}(s)}$,
$g_{\varepsilon}(s)=s^{\frac{1}{2}+\varepsilon}$, and
$\varepsilon\in(0,\min(1,{\color[rgb]{0,0,0}\varepsilon_{1}\beta^{-1}}))$ was
already fixed in (2.5).
First, noting that $f(z)$ is bounded by definition (2.3), then recalling that
$p>3$ (see (1.2)) and $0\leq\chi_{0}(Z)\leq 1$ (see right after (2.4)), we
write

$|\big{(}f(z)+\chi_{0}(Z)\frac{\kappa}{2ps}\big{)}^{p}-f(z)^{p}|\leq\frac{C}{s}.$
(3.14)

Noting that $f^{\prime\prime}(z)$ and $zf^{\prime}(z)$ are also bounded, still
by (2.3), then recalling that $\chi_{0}$ is smooth and compactly supported, we
clearly see from (3.13) that

$|R(y,s)|\leq\frac{C}{s}.$ (3.15)

In particular, by definition (2.16), we have for $s$ large enough:

$|R_{e}(y,s)|\leq\frac{C}{s}\leq\frac{C}{s^{1-\varepsilon_{1}}}.$

Now, we estimate $\|(1+|y|^{\beta})R_{e}(y,s)\|_{L^{\infty}}$. Since we have
the same profile as in [1], arguing as in pages 15-17 of [1], we easily prove
the following estimate

$\|(1+|y|^{\beta})R_{e}(y,s)\|_{L^{\infty}}\leq\frac{C}{s^{1-\frac{\beta}{2}-\varepsilon\beta}}.$

Since $\varepsilon\beta\leq\varepsilon_{1}$ by (2.5), we obtain for $s$ large
enough

$\|(1+|y|^{\beta})R_{e}(y,s)\|_{L^{\infty}}\leq\frac{C}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

Finally, we bound $R_{-}(y,s)$.
Refining the Taylor expansion (3.14) up to the second order, we write

$\displaystyle
R_{i}=\frac{1}{s}(f"(z)+\frac{z}{2}f^{\prime}(z)+\frac{\kappa}{2p}\chi_{0}(Z)f(z)^{p-1})+\frac{p(p-1)}{2}(\frac{\kappa}{2ps})^{2}\chi_{0}^{2}(Z)f^{p-2}(z)+O\left(\frac{1}{s^{3}}\right).$
(3.16)

Since $\chi_{0}$ is constant on the unit ball $B(0,1)$ (see right after
(2.4)), making a Taylor expansion of $R_{i}$ (3.16) in $z$ and in $Z$, we see
that

$R_{i}=\frac{1}{s}\left(a+bz^{2}+O(z^{3})\right)+\frac{c}{s^{2}}(1+O(z^{2}))+O(\frac{1}{s^{3}})+O\left(\frac{|Z|^{3}}{s}\right)$

for some real numbers $a$, $b$ and $c$. Similarly, we have from (3.13)

$R_{ii}=\frac{\kappa}{2ps}\left(\frac{1}{s}-\frac{1}{p-1}\right)+O\left(\frac{|Z|^{3}}{s}\right).$

Using Lemma 3.10 together with straightforward estimates, we see that

$|R_{-}|\leq\frac{C}{s^{\frac{5}{2}}}(1+|y|^{3})\leq\frac{C}{s^{\frac{5}{2}-\varepsilon_{1}}}(1+|y|^{3})$

for $s$ large enough. This concludes the proof of Lemma 3.9. ∎

We now estimate the new term defined in (2.11). For this end, we claim first,
the following Lemma:

######  LEMMA 3.11.

For all $0<\varepsilon_{1}\leq\frac{1}{2}$, $K_{0}\geq 1$ and $A\geq 1$, there
exists $s_{5}(K_{0},A,\varepsilon_{1})$ sufficiently large, such that for all
$s\geq s_{5}(K_{0},A,\varepsilon_{1})$, if
$v\in\mathcal{V}_{A,\varepsilon_{1}}(s)$ satisfies the following

$\|\nabla
v(s)\|_{L^{\infty}}\leq\frac{CA^{2}}{s^{\tau}},\quad\|(1+|y|^{\beta})\nabla
v(s)\|_{L^{\infty}}\leq\frac{CA^{2}}{s^{\tau^{\prime}}},$ (3.17)

for some positive constants $\tau$ and $\tau^{\prime}$, then, the new term
defined in (2.11) satisfies

  1. 1.

$\|\mathcal{N}(y,s)\|_{L^{\infty}}\leq Ce^{-\frac{\gamma}{2}s}$,

  2. 2.

$\|(1+|y|^{\beta})\mathcal{N}_{e}(y,s)\|_{L^{\infty}}\leq
Ce^{-\frac{\gamma}{2}s}$,

where $C$ is a positive constant.

###### Proof.

Since $V_{A,\varepsilon_{1}}$ is increasing (in the sense of inclusion) with
respect to $\varepsilon_{1}$, recalling that
$0<\varepsilon_{1}\leq\frac{1}{2}$ and that the case
$\varepsilon_{1}=\frac{1}{2}$ was treated in Abdelhedi and Zaag [1], we refer
the read to Lemma 3.11 page 18 of [1].
∎

Thanks to Lemma 3.11, we obtain the following results:

######  PROPOSITION 3.12.

For all $0<\varepsilon_{1}\leq\frac{1}{2}$, $K_{0}\geq 1$ and $A\geq 1$, there
exists $s_{6}(K_{0},A,\varepsilon_{1})$ sufficiently large, such that for all
$s\geq s_{6}(K_{0},A,\varepsilon_{1})$, if
$v\in\mathcal{V}_{A,\varepsilon_{1}}(s)$ and $\nabla v$ satisfies the estimate
stated in Proposition 3.7, then, the new term $\mathcal{N}$ defined in (2.11)
satisfies

$|\mathcal{N}_{m}(s)|\leq Ce^{-\frac{\gamma}{2}s},\,0\leq m\leq
2,\quad\displaystyle\|\frac{\mathcal{N}_{-}(y,s)}{1+|y|^{3}}\|_{L^{\infty}}\leq
Ce^{-\frac{\gamma}{2}s}$

and $\mathcal{N}\in\mathcal{V}_{C,\varepsilon_{1}}(s)$, for some positive
constant $C$.

###### Proof.

The proof follows by standard estimates exactly as in the case
$\varepsilon_{1}=\frac{1}{2}$ treated in [1]. See Proposition 3.10 page 18 in
that paper. ∎

####  3.3.2  Step 2: Reduction to a finite dimensional problem

In this subsection, we reduce the problem to a finite dimensional one. We
prove through a priori estimates that the control of $v(s)$ in
$\mathcal{V}_{A,\varepsilon_{1}}(s)$ can be reduced to the control of
$(v_{0},v_{1})(s)$ in $[-\frac{A}{s^{2}},\frac{A}{s^{2}}]^{2}$.
For this end, we recall the integral equation (2.12), for each
$s\geq\sigma\geq s_{0}$ :

$\displaystyle v(s)$ $\displaystyle=$ $\displaystyle
K(s,\sigma)v(\sigma)+\int_{\sigma}^{s}K(s,t)(B(v(t))+R(t)+\mathcal{N}(t))dt$
(3.18) $\displaystyle=$
$\displaystyle\mathcal{A}(s)+\mathcal{B}(s)+\mathcal{C}(s)+\mathcal{D}(s),$

and we estimate the different components of this Duhamel formulation.
In the first step, we recall from [1] this fundamental property concerning the
kernel $K(s,\sigma)$:

######  LEMMA 3.13.

There exists $K_{5}\geq 1$ such that for all $K_{0}\geq K_{5}$, for all
$\rho>0$, there exists $\sigma_{0}=\sigma_{0}(K_{0},\rho)$ such that if
$\sigma\geq\sigma_{0}\geq 1$ and $g(\sigma)$ satisfies

$\displaystyle\sum_{m=0}^{2}|g_{m}(\sigma)|+\|\frac{g_{-}(y,\sigma)}{1+|y|^{3}}\|_{L^{\infty}}+\|(1+|y|^{\beta})g_{e}(\sigma)\|_{L^{\infty}}<+\infty,$
(3.19)

then $\theta(s)=K(s,\sigma)g(\sigma)$ satisfies for all
$s\in[\sigma,\sigma+\rho]$,

  1. 1.

$\displaystyle\displaystyle\|\frac{\theta_{-}(y,s)}{1+|y|^{3}}\|_{L^{\infty}}$
$\displaystyle\leq$ $\displaystyle
C\frac{e^{s-\sigma}((s-\sigma)^{2}+1)}{s}(|g_{0}(\sigma)|+|g_{1}(\sigma)|+\sqrt{s}|g_{2}(\sigma)|)$
$\displaystyle+Ce^{-\frac{s-\sigma}{2}}\|\frac{g_{-}(y,\sigma)}{1+|y|^{3}}\|_{L^{\infty}}+C\frac{e^{-(s-\sigma)^{2}}}{s^{\frac{3}{2}}}\|g_{e}(\sigma)\|_{L^{\infty}}$

  2. 2.

$|\theta_{e}(s)|\leq\displaystyle
Ce^{s-\sigma}\big{(}\sum_{l=0}^{2}s^{\frac{l}{2}}|g_{l}(\sigma)|+s^{\frac{3}{2}}\|\frac{g_{-}(y,\sigma)}{1+|y|^{3}}\|_{L^{\infty}}\big{)}+Ce^{-\frac{s-\sigma}{p}}\|g_{e}(\sigma)\|_{L^{\infty}}.$

  3. 3.

$\displaystyle\displaystyle\|(1+|y|^{\beta})\theta_{e}(y,s)\|_{L^{\infty}}$
$\displaystyle\leq$ $\displaystyle\displaystyle
Ce^{-\frac{s-\sigma}{2}(\frac{1}{p-1}-\frac{\beta}{2})}\|(1+|y|^{\beta})g_{e}(y,\sigma)\|_{L^{\infty}}$
$\displaystyle+\displaystyle
Ce^{\frac{\beta}{2}(s-\sigma)}s^{\frac{\beta}{2}}(\sum_{l=0}^{2}s^{\frac{l}{2}}|g_{l}(\sigma)|+s^{\frac{3}{2}}\|\frac{g_{-}(y,\sigma)}{1+|y|^{3}}\|_{L^{\infty}}).$

###### Proof.

The original version of this lemma is due to Bricmont and Kupiainen in [5],
where they gave the proof in the case where some particular bounds hold on
each component appearing in 3.19, only for $\beta=0$. Later in [31], Nguyen
and Zaag simply restated the lemma of Bricmont and Kupiainen and checked that
the proof of [5] works without those particular bounds, still for $\beta=0$.
In our earlier paper [1], we handled the case $0\leq\beta<\frac{2}{p-1}$.
Accordingly, we refer the reader to Lemma 3.15 page 26 in [1] for the detailed
justification. ∎

Applying the above Lemma, we get a new bound on all terms in the decomposition
$(\ref{decomp1})$. More precisely, we have the following results:

######  LEMMA 3.14.

For all $0<\varepsilon_{1}\leq\frac{1}{2}$, there exists
$K_{6}(\varepsilon_{1})$ such that for any $K_{0}\geq K_{6}(\varepsilon_{1})$,
there exists $A_{6}(K_{0},\varepsilon_{1})>0$ such that for all $A\geq
A_{6}(K_{0},\varepsilon_{1})$ and $\rho>0$, there exists
$s_{06}(K_{0},A,\rho,\varepsilon_{1})>0$ with the following property: for all
$s_{0}\geq s_{06}(K_{0},A,\rho,\varepsilon_{1})$ assume that for all
$s\in[\sigma,\sigma+\rho]$, $v(s)$ satisfies $(\ref{eq_v})$,
$v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$ and $\nabla v$ satisfies the
estimates stated in Proposition 3.7. Then, we have:

  1. 1.

Linear term:

$\|\frac{\mathcal{A}-(y,s)}{1+|y|^{3}}\|_{L^{\infty}}\leq
Ce^{-\frac{1}{2}(s-\sigma)}\|\frac{v_{-}(y,\sigma)}{1+|y|^{3}}\|_{L^{\infty}}+C\frac{e^{-(s-\sigma)^{2}}}{s^{\frac{3}{2}}}\|v_{e}(\sigma)\|_{L^{\infty}}+\frac{C}{s^{\frac{5}{2}-\varepsilon_{1}}}.$

$\|\mathcal{A}_{e}\|_{L^{\infty}}\leq
Ce^{-\frac{s-\sigma}{p}}\|v_{e}(\sigma)\|_{L^{\infty}}+Ce^{s-\sigma}s^{\frac{3}{2}}\|\frac{v_{-}(y,\sigma)}{1+|y|^{3}}\|_{L^{\infty}}+\frac{C}{s^{1-\varepsilon_{1}}}.$
$\displaystyle\|(1+|y|^{\beta})\mathcal{A}_{e}(s)\|_{L^{\infty}}\\!\\!\\!\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!Ce^{-\frac{s-\sigma}{2}(\frac{1}{p-1}-\frac{\beta}{2})}\|(1+|y|^{\beta})v_{e}(\sigma)\|_{L^{\infty}}\\!+\\!Ce^{\frac{\beta}{2}(s-\sigma)}s^{\frac{\beta}{2}+\frac{3}{2}}\|\frac{v_{-}(y,s)}{1+|y|^{3}}\|_{L^{\infty}}$
$\displaystyle+$
$\displaystyle\frac{C}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

  2. 2.

Nonlinear term:

$\|\frac{\mathcal{B}_{-}(y,s)}{1+|y|^{3}}\|_{L^{\infty}}\leq\frac{C}{s^{\frac{5}{2}-\varepsilon_{1}}}(s-\sigma),\quad\|\mathcal{B}_{e}(s)\|_{L^{\infty}}\leq\frac{C}{s^{1-\varepsilon_{1}}}(s-\sigma)e^{s-\sigma}.$
$\|(1+|y|^{\beta})\mathcal{B}_{e}(s)\|_{L^{\infty}}\leq
C\frac{1+e^{\frac{\beta}{2}(s-\sigma)}(s-\sigma)}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

  3. 3.

Remainder term:

$\|\frac{\mathcal{C}_{-}(y,s)}{1+|y|^{3}}\|_{L^{\infty}}\leq\frac{C}{s^{\frac{5}{2}-\varepsilon_{1}}}(s-\sigma),\quad\|\mathcal{C}_{e}(s)\|_{L^{\infty}}\leq\frac{C}{s^{1-\varepsilon_{1}}}(s-\sigma)e^{s-\sigma}.$
$\|(1+|y|^{\beta})\mathcal{C}_{e}(s)\|_{L^{\infty}}\leq
C\frac{1+e^{\frac{\beta}{2}(s-\sigma)}(s-\sigma)}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

  4. 4.

New term:

$\|\frac{\mathcal{D}_{-}(y,s)}{1+|y|^{3}}\|_{L^{\infty}}\leq
C(s-\sigma)e^{s-\sigma}e^{-\frac{\gamma}{2}s},\quad\|\mathcal{D}_{e}(s)\|_{L^{\infty}}\leq
C(s-\sigma)e^{s-\sigma}e^{-\frac{\gamma}{2}s}.$
$\|(1+|y|^{\beta})\mathcal{D}_{e}(s)\|_{L^{\infty}}\leq
C(s-\sigma)e^{-\frac{\gamma}{2}s}(1+s^{\frac{\beta}{2}}e^{\frac{\beta}{2}(s-\sigma)}).$

###### Proof.

Although we have a different shrinking set here, in comparison with our
earlier work [1], the proof follows exactly in the same way. See Lemma 3.16
page 30 in that paper for details. ∎

In the following statement, we add the bounds of Lemma 3.14 in order to obtain
bounds on $v(y,s)$, thanks to the expression (3.18). We also project equation
(2.7) in order to write differential inequalities satisfies by $v_{m}$ for
$m=0$, $1$ and $2$:

######  PROPOSITION 3.15.

For all $0<\varepsilon_{1}\leq\frac{1}{2}$, there exists
$K_{7}(\varepsilon_{1})$ such that for any $K_{0}\geq K_{7}(\varepsilon_{1})$,
there exists $A_{7}(K_{0},\varepsilon_{1})\geq 1$ such that for all $A\geq
A_{7}(K_{0},\varepsilon_{1})$, there exists $s_{07}(K_{0},A,\varepsilon_{1})$
large enough such that the following holds for all $s_{0}\geq
s_{07}(K_{0},A,\varepsilon_{1})$:
Assume that for some $s_{1}\geq\sigma\geq s_{0}$, we have

$v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s),\quad\mbox{for
all}\;s\in[\sigma,s_{1}],$

and that $\nabla v$ satisfies the estimates stated in Proposition 3.7. Then,
the following holds for all $s\in[\sigma,s_{1}]$,

  1. i)

(ODE satisfied by the positive modes) : For $m\in\\{0,1\\}$, we have

$\displaystyle|v^{\prime}_{m}(s)-(1-\frac{m}{2})v_{m}(s)|\leq\frac{C}{s^{2}}.$

  2. ii)

(ODE satisfied by the null mode) : We have

$|v^{\prime}_{2}(s)+\frac{2}{s}v_{2}(s)|\leq\frac{C}{s^{3}}.$

  3. iii)

(Control of the negative and outer modes): We have

$\displaystyle\displaystyle\|\frac{v_{-}(s)}{1+|y|^{3}}\|_{L^{\infty}}\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!Ce^{-\frac{s-\sigma}{2}}\|\frac{v_{-}(\sigma)}{1+|y|^{3}}\|_{L^{\infty}}\\!+\\!C\frac{e^{-(s-\sigma)^{2}}}{s^{\frac{3}{2}}}\|v_{e}(\sigma)\|_{L^{\infty}}\\!+\\!C\frac{1+s-\sigma}{s^{\frac{5}{2}-\varepsilon_{1}}},$
$\displaystyle\|v_{e}(s)\|_{L^{\infty}}\\!\\!\\!\\!\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!\\!Ce^{-\frac{s-\sigma}{p}}\|v_{e}(\sigma)\|_{L^{\infty}}\\!+\\!Ce^{s-\sigma}s^{\frac{3}{2}}\|\frac{v_{-}(\sigma)}{1+|y|^{3}}\|_{L^{\infty}}\\!+\\!C\frac{1+(s-\sigma)e^{s-\sigma}}{s^{1-\varepsilon_{1}}}.$

  4. iv)

(Weighted estimate of the outer mode):

$\displaystyle\displaystyle\|(1+|y|^{\beta})v_{e}(s)\|_{L^{\infty}}\\!\\!\\!$
$\displaystyle\leq$ $\displaystyle\\!\\!\\!C\displaystyle
e^{-\frac{s-\sigma}{2}(\frac{1}{p-1}-\frac{\beta}{2})}\|(1+|y|^{\beta})v_{e}(\sigma)\|_{L^{\infty}}$
$\displaystyle+$ $\displaystyle
Ce^{\frac{\beta}{2}(s-\sigma)}s^{\frac{3}{2}+\frac{\beta}{2}}\|\frac{v_{-}(\sigma)}{1+|y|^{3}}\|_{L^{\infty}}+C\frac{1+(s-\sigma)e^{\frac{\beta}{2}(s-\sigma)}}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

###### Proof.

The proof of items iii) and iv) is straightforward from Lemma 3.14 and the
decomposition (3.18).
As for items i) and ii), they follow exactly in the same way as in any paper
about the standard heat equation ($\mu=0$), since the effect of the “new term”
is exponentially small, thanks to Lemma 3.11. See for example Lemma 3.10 page
1293 of Nguyen and Zaag [29]. ∎

Now, with Proposition 3.15, which gives estimates on the different components
of the decomposition (3.18), the proof of Proposition 3.8 follows exactly as
in our earlier work [1]. For that reason, we omit the proof and refer the
reader to page 31 of that paper, where the analogous statement (numbered
Proposition 3.5) is proved.

###  3.4 Proof of the existence of a solution in
$\mathcal{V}_{A,\varepsilon_{1}}(s)$

In this subsection, we solve the two-dimensional problem using index theory
and we prove the existence of a solution $v$ of (2.7) in
$\mathcal{V}_{A,\varepsilon_{1}}(s)$. More precisely, we have the following
statement:

######  PROPOSITION 3.16 (Existence of a solution in the shrinking set).

For all $0<\varepsilon_{1}\leq\frac{1}{2}$, there exists
$K_{4}(\varepsilon_{1})\geq 1$ such that for all $K_{0}\geq
K_{4}(\varepsilon_{1})$, there exists $A_{4}(K_{0},\varepsilon_{1})\geq 1$
such that for all $A\geq A_{4}(K_{0},\varepsilon_{1})$ there exists
$s_{04}(K_{0},A,\varepsilon_{1})$ such that for all $s_{0}\geq
s_{04}(K_{0},A,\varepsilon_{1})$, there exists $(d_{0},d_{1})$ such that if
$v$ is the solution of $(\ref{eq_v})$ with initial data at $s_{0}$, given in
$(\ref{ci})$, then $v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$, for all $s\geq
s_{0}$.

###### Proof.

The derivation of this proposition from the reduction to a finite dimensional
problem and the transverse crossing (both given in Proposition 3.8) is
classical in the literature concerning construction of solutions with
prescribed behavior. For that reason, we omit it and refer the reader to page
13 of [1] where an analogous statement is given (namely, Proposition 3.7). The
proof in that paper holds here with natural modifications. ∎

##  4 Proof of the main result

This section is dedicated to the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Consider some $\varepsilon_{1}\in(0,\frac{1}{2}]$ and
$\alpha\in(0,\frac{1}{2})$. Using Proposition 3.16, we derive the existence of
a solution $v$ of equation $(\ref{eq_v})$ defined for all $y\in\mathbb{R}$ and
$s\geq s_{0}$, such that $v(s)\in\mathcal{V}_{A,\varepsilon_{1}}(s)$ for some
$s_{0}\geq 1$ and $A>0$. Moreover, using (3.12), we have for all $s\geq
s_{0}$,

$\|(1+|y|^{\beta})v(s)\|_{L^{\infty}}+\|(1+|y|^{\beta})\nabla
v(s)\|_{L^{\infty}}\leq\displaystyle\frac{C(A)}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}$

where $C(A)=CA^{2}$. For simplicity, we omit the dependence on $A$ in the
following.
By definition (2.6) of $v$, we see that for all $s\geq s_{0}$

$\|(1+|y|^{\beta})\big{(}w(y,s)-\varphi(y,s)\big{)}\|_{L^{\infty}}+\|(1+|y|^{\beta})\nabla_{y}\big{(}w(y,s)-\varphi(y,s)\big{)}\|_{L^{\infty}}\leq\displaystyle\frac{C}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}},$

where $\varphi$ is the profile introduced in $(\ref{n_profile})$.
First, we estimate this profile. We remark that our ”intermediate” profile
satisfies:

$|\displaystyle
f(\frac{y}{\sqrt{s}})|\sim\big{(}\frac{s}{b|y|^{2}}\big{)}^{\frac{1}{p-1}}\;\mbox{as}\;\frac{y}{\sqrt{s}}\to+\infty,$

and for all $K_{0}\geq 1$ and $|y|\geq 2K_{0}\sqrt{s}$, we have

$\displaystyle|\frac{1}{\sqrt{s}}\nabla
f(\frac{y}{\sqrt{s}})|\leq\frac{C}{(1+|y|^{\beta})s^{\frac{1}{2}-\frac{\beta}{2}}},$

on the one hand (remember that $\beta<2/(p-1)$).
On the other hand, the cut-off function $\chi_{0}$ satisfies for all
$y\in\mathbb{R}$,

$\displaystyle|\frac{\kappa
N}{2ps}\chi_{0}(\frac{y}{g_{\varepsilon}(s)})|\leq\frac{C}{(1+|y|^{\beta})s^{1-\frac{\beta}{2}-\varepsilon\beta}},\quad|\frac{\kappa
N}{2ps}\frac{1}{g_{\varepsilon}(s)}\nabla\chi_{0}(\frac{y}{g_{\varepsilon}(s)})|\leq\frac{C}{(1+|y|^{\beta})s^{\frac{3}{2}-\frac{\beta}{2}-\varepsilon(\beta-1)}}.$

With the choice of $\varepsilon\in(0,\min(1,\varepsilon_{1}\beta^{-1}))$ (see
(2.5)), we obtain

$\displaystyle|\frac{\kappa
N}{2ps}\chi_{0}(\frac{y}{g_{\varepsilon}(s)})|\leq\frac{C}{(1+|y|^{\beta})s^{1-\frac{\beta}{2}-\varepsilon_{1}}},\quad|\frac{\kappa
N}{2ps}\frac{1}{g_{\varepsilon}(s)}\nabla\chi_{0}(\frac{y}{g_{\varepsilon}(s)})|\leq\frac{C}{(1+|y|^{\beta})s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

Using the above estimates, we deduce that

$\|(1+|y|^{\beta})(w(y,s)-f(\frac{y}{\sqrt{s}}))\|_{L^{\infty}}+\|(1+|y|^{\beta})\nabla_{y}(w(y,s)-f(\frac{y}{\sqrt{s}}))\|_{L^{\infty}}\leq\displaystyle\frac{C}{s^{1-\frac{\beta}{2}-\varepsilon_{1}}}.$

Therefore, thanks to the similarity variables transformation (2.1), the
solution $u$ of equation $(\ref{eq_u})$ defined for all $x\in\mathbb{R}$ and
$t\in[0,T)$ satisfies

$|(T-t)^{\frac{1}{p-1}}u(x,t)-f(\frac{x}{\sqrt{(T-t)|\log(T-t)|}})|\leq\displaystyle\frac{C}{(1+(\frac{|x|^{2}}{T-t})^{\frac{\beta}{2}})|\log(T-t)|^{1-\frac{\beta}{2}-\varepsilon_{1}}},$

and

$\displaystyle|(T-t)^{\frac{1}{2}+\frac{1}{p-1}}\nabla u(x,t)$
$\displaystyle-$ $\displaystyle\frac{1}{\sqrt{|\log(T-t)|}}\nabla
f(\frac{x}{\sqrt{(T-t)|\log(T-t)|}})|$
$\displaystyle\leq\displaystyle\frac{C}{(1+(\frac{|x|^{2}}{T-t})^{\frac{\beta}{2}})|\log(T-t)|^{1-\frac{\beta}{2}-\varepsilon_{1}}},$

and identities (1.13) and (1.14) follow. In particular, $\lim_{t\to
T}(T-t)^{\frac{1}{p-1}}u(0,t)=(p-1)^{\frac{1}{p-1}}$ and $u$ blows up at time
$t=T$ at the origin.

Since our results in item $1)$ of this Theorem are better that our earlier
result in [1], the single point blow-up property for $u$ and $\nabla u$
follows exactly as in our paper [2]. If for $u$, we clearly have from (1.13)
that $u(0,t)\to\infty$ as $t\to T$, the question to find some $\xi(t)\to 0$
such that $|\nabla u(\xi(t),t)|\to 0$ was left open after our previous work.
In this new work, we are able to prove it, as we have already stated in the
third remark following the statement of Theorem 1.1, and as we show in the
following: If we choose $\varepsilon_{1}$ such that
$\varepsilon_{1}\leq\frac{1}{2}-\frac{\beta}{2}$, then we have

$(T-t)^{\frac{1}{p-1}}\sqrt{(T-t)|\log(T-t)|}\nabla
u(\sqrt{(T-t)|\log(T-t)|},t)\sim\nabla f(K_{0})\;\mbox{ as}\;t\to T.$

In particular,

$\|\nabla
u(t)\|_{L^{\infty}}\geq\frac{C}{(T-t)^{\frac{1}{p-1}}\sqrt{(T-t)|\log(T-t)|}}.$

Thus $\nabla u$ blows up at time $t=T$ at the origin. This concludes the proof
of parts $1)$ and $2)$ of Theorem 1.1.

Finally, for the proof of part $3)$ of Theorem 1.1, the reader can adapt
easily the proof in [2] (section $4$ pages 2618-2622), where we replace the
cut off function in [2] (step 1 page 2619), by
$\phi_{r}(\xi)=\phi(\displaystyle\frac{\xi}{r|\log(T-t_{0})|^{\alpha}})$,
$0<\alpha<\frac{1}{2}$ and the bounded in estimate $(4.1)$ in [2] page 2618,
by $\displaystyle\frac{c}{|\log(T-t_{0})|^{1-\varepsilon_{1}}}$,
$0<\varepsilon_{1}\leq\frac{1}{2}$. ∎

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pdftexdestination with the same hyperrefOption hyperrefToken pdftex(dest)

# Overestimated inclinations of Milgromian disc galaxies: the case of the
ultradiffuse galaxy AGC 114905

Indranil Banik1, Srikanth. T. Nagesh2,3, Hosein Haghi3,4, Pavel Kroupa3,5 and
Hongsheng Zhao1
1Scottish Universities Physics Alliance, University of Saint Andrews, North
Haugh, Saint Andrews, Fife, KY16 9SS, UK
2Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
D-53121 Bonn, Germany
3Helmholtz-Institut für Strahlen und Kernphysik (HISKP), University of Bonn,
Nussallee 14$-$16, D-53115 Bonn, Germany
4Department of Physics, Institute for Advanced Studies in Basic Sciences
(IASBS), PO Box 11365-9161, Zanjan, Iran
5Astronomical Institute, Faculty of Mathematics and Physics, Charles
University, V Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic Email:
<EMAIL_ADDRESS>(Indranil Banik),
<EMAIL_ADDRESS>(Srikanth T. Nagesh)

###### Abstract

We present hydrodynamical star-forming simulations in the Milgromian dynamics
(MOND) framework of a gas-rich disc galaxy with properties similar to AGC
114905, which has recently been argued to have a rotation curve (RC) that is
inconsistent with the MOND prediction. Our first model considers the galaxy in
isolation, while our second model includes an external field of
$0.05\,a_{{}_{0}}$, the estimated gravitational field from large-scale
structure. We show that isophotes in the face-on view can differ from circular
at the 50% level. This could mislead observers into overestimating the
inclination $i$ between disc and sky planes. Because RCs require a correction
factor of $1/\sin i$, the actual RC could be much higher than reported by
observers. This plausibly reconciles AGC 114905 with MOND expectations.

###### keywords:

gravitation – galaxies: disc – galaxies: dwarf – galaxies: fundamental
parameters – galaxies: individual: AGC 114905 – galaxies: kinematics and
dynamics

††pubyear: 2022††pagerange: Overestimated inclinations of Milgromian disc
galaxies: the case of the ultradiffuse galaxy AGC 114905–References

##  1 Introduction

The observed rotation curves (RCs) of galaxies are flat in their outskirts
(e.g. Faber & Gallagher, 1979, and references therein). This strongly
contradicts the expected Keplerian decline beyond the extent of the luminous
matter, where the Newtonian inverse square law implies that the circular
rotation velocity $v_{c}$ scales with galactocentric radius $r$ as
$v_{c}\propto 1/\sqrt{r}$. This acceleration discrepancy could indicate that
galaxies are embedded in massive extended haloes of cold dark matter (CDM;
Ostriker & Peebles, 1973). These haloes are nowadays a cornerstone of the
standard cosmological framework known as $\Lambda$CDM (Efstathiou, Sutherland
& Maddox, 1990; Ostriker & Steinhardt, 1995). However, several very serious
problems have been identified with $\Lambda$CDM on galactic scales (Peebles &
Nusser, 2010; Kroupa et al., 2010; Kroupa, 2012, 2015; Bullock & Boylan-
Kolchin, 2017; Bose et al., 2019; Haslbauer et al., 2019; Pawlowski, 2021). It
also faces serious problems on cosmological scales, including the Hubble
tension (Di Valentino, 2021) but also with regards to voids (Haslbauer, Banik
& Kroupa, 2020) and galaxy clusters (Asencio, Banik & Kroupa, 2021) being more
extreme than predicted in this framework.

$\Lambda$CDM relies on the standard law of gravitation, which was formulated
using only Solar System data. Rather than postulating the existence of exotic
CDM particles beyond the well-tested standard model of particle physics,
Milgrom (1983) noted that the acceleration discrepancies arise when the
gravitational field $g$ is much weaker than in the Solar System. He introduced
a new constant of nature ($a_{{}_{0}}$) with dimensions of an acceleration
such that in an isolated spherically symmetric system with Newtonian gravity
$g_{{}_{N}}$,

$\displaystyle
g~{}=~{}\left\\{\begin{array}[]{ll}\sqrt{g_{{}_{N}}a_{{}_{0}}}&\quad\left(g_{{}_{N}}\ll
a_{{}_{0}}\right)\,,\\\ g_{{}_{N}}&\quad\left(g_{{}_{N}}\gg
a_{{}_{0}}\right)\,.\end{array}\right.$ (3)

This Milgromian dynamics (MOND) theory is one of the most competitive
alternative solutions to the acceleration discrepancy problem, as extensively
reviewed in Famaey & McGaugh (2012). Empirically, $a_{{}_{0}}=1.2\times
10^{-10}$ m/s2 or 3.9 pc/Myr2 (Begeman, Broeils & Sanders, 1991), with more
recent studies confirming this value (Gentile, Famaey & de Blok, 2011;
McGaugh, Lelli & Schombert, 2016). In the deep-MOND limit, MOND satisfies a
principle known as spacetime scale invariance according to which any
trajectory $\left(t,\bm{r}\right)$ that solves the equations of motion implies
that these are also solved by another trajectory $\left(\lambda
t,\lambda\bm{r}\right)$ for any constant $\lambda$ (Milgrom, 2009).
Considering the case of uniform circular motion reveals that the rotation
curve of an isolated point mass must be flat in the deep-MOND regime.
Newtonian dynamics does not obey spacetime scale invariance.

The limits in Equation 3 follow from a new non-relativistic gravity theory
based on a Lagrangian, thereby respecting the usual conservation laws with
respect to linear and angular momentum and the energy (Bekenstein & Milgrom,
1984). The generalized non-linear Poisson equation derived from this
Lagrangian leads to a rich behaviour of gravitating systems, accounting
automatically for hitherto unexplained phenomena like the KBC void and Hubble
tension (Haslbauer et al., 2020), extreme galaxy cluster collisions at high
redshift (Katz et al., 2013; Asencio et al., 2021), the Local Group satellite
planes (Banik et al., 2018, 2022; Bílek et al., 2018, 2021), and many others
besides while remaining consistent with galaxy cluster dynamics and early
Universe observations (for an extensive review, see Banik & Zhao, 2022). The
MOND Poisson equation can be solved using standard mesh-refinement codes like
phantom of ramses (por; Lüghausen et al., 2015; Nagesh et al., 2021) and
raymond (Candlish et al., 2015).

A robust prediction of the MOND Poisson equation is that galaxy RCs become
asymptotically flat at some velocity $v_{{}_{f}}$ which depends on the
baryonic mass $M_{b}$ according to

$\displaystyle v_{{}_{f}}~{}=~{}\left(GM_{b}a_{{}_{0}}\right)^{1/4}\,.$ (4)

More generally, MOND predicts a very tight radial acceleration relation (RAR)
between $g$ as deduced from RCs and $g_{{}_{N}}$ as deduced from the
distribution of observable matter. A tight RAR is indeed apparent
observationally (McGaugh et al., 2016; Lelli et al., 2017) from the Spitzer
Photometry and Accurate Rotation Curves (SPARC) dataset (Lelli, McGaugh &
Schombert, 2016). RCs offer strong tests of the MOND paradigm, especially if
the central surface density $\Sigma_{0}\ll\Sigma_{M}$, where the critical
surface density in MOND is

$\displaystyle\Sigma_{M}~{}\equiv~{}\frac{a_{{}_{0}}}{2\mathrm{\pi}G}~{}=~{}137\,M_{\odot}/\text{pc}^{2}\,.$
(5)

This is because $g\gg g{{}_{N}}$ in such low surface brightness (LSB)
galaxies, making them well suited for testing different explanations for the
galactic acceleration discrepancies (McGaugh, 2020).

In a recent paper, Mancera Piña et al. (2022) report new HI interferometric
observations of the gas-rich ultra-diffuse galaxy AGC 114905, tracing its HI
emission up to 10 kpc from the galaxy centre. They infer from these a
surprisingly low $v_{{}_{f}}\approx 23$ km/s. They then deduce that the
observed RC of AGC 114905 can be explained almost entirely by $g_{{}_{N}}$ of
the baryons alone, with little room for CDM within the outermost observed
radius. They find that the RC can only be reproduced by standard CDM haloes if
the halo concentration is as low as 0.3, completely off $\Lambda$CDM
expectations. They also argue that the observed RC significantly disagrees
with the normalization and shape of the MOND RC.

In this paper, we use fully self-consistent hydrodynamical MOND simulations
that include star formation and supernova feedback to show that LSB Milgromian
disc galaxies are often not round when viewed face-on. This can lead to the
inclination $i$ between disc and sky planes appearing to be much higher than
it actually is. This is critical because the claim of Mancera Piña et al.
(2022) to have falsified MOND relies on the correctness of their measured
$i=32^{\circ}$, but those authors showed that consistency is gained for an
almost face-on $i=11^{\circ}$. We argue that this is quite possible, so the
kinematics of AGC 114905 are consistent with MOND. Moreover, even a small
amount of warping in such a nearly face-on galaxy could seriously impact the
shape of the RC deduced from line-of-sight velocities.

This paper is structured as follows: In Section 2, we describe the simulations
and how we analyse them. We then present our results in Section 3 and discuss
their implications in Section 4. Our conclusions are given in Section 5.

##  2 Methods

###  2.1 MOND galaxy simulations

Parameter | Value
---|---
Total mass | $1.42\times 10^{9}M_{\odot}$
Stellar mass | $1.30\times 10^{8}M_{\odot}$
Gas fraction | 0.90
Stellar disc scale length | 1.79 kpc
Gas disc scale length | 4.47 kpc
$\Sigma_{0}/\Sigma_{M}$ | 0.122
$v_{{}_{f}}$ | 68.96 km/s
Best resolution | 195 pc
Table 1: Parameters of our galaxy models. The isolated model assumes
$g_{\textrm{ext}}=0$, while the EFE model assumes
$g_{\textrm{ext}}=0.05\,a_{{}_{0}}$ directed at $30^{\circ}$ to the disc plane
but with no component along $y$.

The galaxy models presented here were evolved for 5 Gyr. They have two main
components: an inner stellar disc with scale length $r_{d}$ and an outer gas
disc with scale length $r_{g}=2.5\,r_{d}$ (this factor will be justified in a
future work; see S. T. Nagesh et al. 2022, in preparation). Both discs have an
exponential surface density profile initially (Freeman, 1970). Their adopted
parameters (listed in Table 1) were chosen based on the properties of AGC
114905 discussed in section 2 of Mancera Piña et al. (2022). The models are
entirely in the MOND regime as the central surface density $\Sigma_{0}$ is
below the MOND surface density $\Sigma_{M}$ (Equation 5): the ratio
$\Sigma_{0}/\Sigma_{M}=0.12$ initially (see Table 1). For reference, this
ratio was 10 in the MOND model presented in Roshan et al. (2021), while the
M33 model presented in Banik et al. (2020) used a lower ratio of $\approx 2$
(see their figure 1). Thus, our work considers a much lower $\Sigma_{0}$ than
these earlier MOND simulations, which is needed to match the properties of AGC
114905. Note that since the equations of motion become self-similar in the
deep-MOND regime due to the spacetime scale invariance discussed in Section 1,
our isolated results are also applicable to galaxies with a much lower
$\Sigma_{0}$. However, in the presence of an external field, its strength
relative to the internal gravity of the disc provides another dimensionless
parameter (Section 2.1.1).

The axisymmetric initial conditions for these galaxy models were generated
using Disk Initial Conditions Environment (dice; Perret et al., 2014). dice
was modified to initialize discs in Milgromian gravity (m-dice; Banik et al.,
2020). m-dice is publicly
available.111https://bitbucket.org/SrikanthTN/bonnPoR/src/master/ We use the
hydrodynamic version of this (h-dice; Nagesh et al., 2021), with the gas
temperature settings being the same as in Banik et al. (2020).

These disc galaxy templates were advanced using por, which numerically
implements the quasi-linear formulation of MOND (QUMOND; Milgrom, 2010) by a
patch to the 2015 version of the hydrodynamics and $N$-body solver ramses
(Teyssier, 2002). The governing equation of QUMOND is:

$\displaystyle\nabla\cdot\bm{g}~{}=~{}\nabla\cdot\left(\nu\bm{g}_{{}_{N}}\right)\,,\quad\nu~{}=~{}\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{a_{{}_{0}}}{g_{{}_{N}}}}\,.$
(6)

We transition between the asymptotic limits in Equation 3 using the simple
interpolating function as this works well observationally (Famaey & Binney,
2005; Iocco et al., 2015; Lelli et al., 2017; Banik & Zhao, 2018). Notice that
$\bm{g}_{{}_{N}}$ first has to be determined from the baryonic distribution
using standard techniques, allowing the algorithm to find $\nu$ at every point
on the grid and thereby algebraically determine the source term for $\bm{g}$.
This is found subject to the boundary condition that the potential $\Phi$ is
asymptotically given by:

$\displaystyle\Phi~{}=~{}\sqrt{GMa_{{}_{0}}}\ln R\,,$ (7)

where $M$ is the total mass inside the simulation volume and $R$ is the
distance from the barycentre in the simulation unit of length, the choice of
which has no bearing on the results. Only a standard Poisson solver is
required, greatly reducing the computational difficulties compared to the
earlier version of MOND (Bekenstein & Milgrom, 1984).

por is publicly available${}^{\ref{PoRbit}}$ along with a user manual to set
up isolated and interacting disc galaxy simulations and extract results from
them into human-readable form (Nagesh et al., 2021). We extracted the particle
and gas data from the outputs at 100 Myr intervals using extract_por and
rdramses, respectively. Both codes are publicly available.${}^{\ref{PoRbit}}$

We consider the face-on view of each galaxy for a distant observer along the
$z$-axis. To do this, we first find the combined barycentre of the stars and
gas. We subtract its position and velocity from the gas cell data and then bin
the gas cells according to their central $\left(x,y\right)$. In this way, we
build up a 2D distribution of the gas surface density $\Sigma_{\text{gas}}$.
We then plot contours on this image and analyse a representative contour in
the outer region, as described next. The level of this outer contour is chosen
so its size is similar to the outermost isophote used by Mancera Piña et al.
(2022) to estimate $i$ (see their figure 7).

####  2.1.1 Including the external field effect (EFE)

The non-linearity of MOND gives rise to the so-called EFE, which causes the
self-gravity of a system to be weakened if the system as a whole is
accelerated uniformly by an external field $\bm{g}_{\textrm{ext}}$, even in
the complete absence of tidal effects. This can be understood using Equation
6: the argument of $\nu$ includes the Newtonian gravity due to both the system
and external sources beyond it. The EFE is a fundamental consequence of MOND
that has always been part and parcel of it (see section 3 of Milgrom, 1983).
Observational evidence for the EFE is discussed further in section 3.3 of
Banik & Zhao (2022). Disc galaxies experiencing the strong EFE of a cluster
environment can become lopsided (Candlish et al., 2018), though even a rather
weak EFE can have subtle consequences (Banik et al., 2020).

We include the EFE in our simulations using the technique discussed in Banik
et al. (2020). As in that work, we assume $\bm{g}_{{}_{\textrm{ext}}}$ lies
entirely within the $xz$ plane and is directed $30^{\circ}$ above the disc
plane. We take $g_{{}_{\textrm{ext}}}=0.05\,a_{{}_{0}}$ since this is roughly
the preferred value for the EFE on very large scales in a MOND model of the
local supervoid which solves the Hubble tension (Haslbauer et al., 2020).
Thus, even a very isolated galaxy should experience an external field of a few
percent of $a_{{}_{0}}$. If the galaxy has a very low surface density such
that $\Sigma_{0}/\Sigma_{M}\ll 0.05$, then the EFE would dominate everywhere,
making the galaxy quasi-Newtonian (see section 2.4 of Banik & Zhao, 2022).
Since the equations of motion in this regime are like those of Newtonian
dynamics but with renormalized $G$, the disc should be unstable for much the
same reason that a bare disc is unstable in Newtonian gravity (Hohl, 1971).
Thus, MOND could struggle to explain a thin disc with a sufficiently low
surface brightness, though this would depend on the estimated
$g_{\textrm{ext}}$.

In our EFE model, the barycentre drifts due to numerical effects (see footnote
14 of Banik et al., 2020). We found that the position of the barycentre
changes very nearly quadratically with time, indicating an acceleration of
$1.6\times 10^{-3}\,a_{{}_{0}}$ in a direction almost opposite to that of
$\bm{g}_{\textrm{ext}}$ and thus almost entirely within the $xz$ plane. This
is expected in a problem that is symmetric with respect to $\pm y$. The
barycentre drift can be reduced with a larger box, but the resolution would
then be poorer. The only way around this problem is to raise the maximum
number of refinement levels, but this is quite computationally intensive.
Another issue is that the disc precesses by $\approx 11^{\circ}$ by the end of
the simulation, so we rotated the coordinate system in our EFE model such that
the images are the view for an observer located very far away in the direction
of the total angular momentum.

###  2.2 Fitting a contour with a circle and an ellipse

We fit a circle and an ellipse to a representative contour in
$\Sigma_{\text{gas}}$. For this, we analyse the list of
$\left(x_{i},y_{i}\right)$, where the integer $i$ labels the $N$ different
points on the contour. We use an algebraic method to obtain an accurate
initial guess (Section 2.2.1) for an iterative scheme to find the optimal
circle and ellipse parameters (Section 2.2.2).

####  2.2.1 The initial guess for the model parameters

The estimated centre of the circle and ellipse are both at
$\left(\overline{x},\overline{y}\right)$, where

$\displaystyle\overline{x}~{}\equiv~{}\frac{1}{N}\sum_{i}x_{i}\,,\quad\overline{y}~{}\equiv~{}\frac{1}{N}\sum_{i}y_{i}\,.$
(8)

We subtract this from each point on the contour to obtain

$\displaystyle\widetilde{x}_{i}~{}\equiv~{}x_{i}-\overline{x}\,,\quad\widetilde{y}_{i}~{}\equiv~{}y_{i}-\overline{y}\,.$
(9)

These quantities are used to calculate the inertia tensor

$\displaystyle\textbf{{I}}~{}\equiv~{}\begin{bmatrix}S_{xx}&S_{xy}\\\
S_{xy}&S_{yy}\end{bmatrix}\,,\quad\text{where}$ (10) $\displaystyle
S_{xx}~{}\equiv~{}\sum_{i}\widetilde{x}_{i}\widetilde{x}_{i}\,,~{}S_{xy}~{}\equiv~{}\sum_{i}\widetilde{x}_{i}\widetilde{y}_{i}\,,~{}S_{yy}~{}\equiv~{}\sum_{i}\widetilde{y}_{i}\widetilde{y}_{i}\,.$

We then diagonalize I to find its eigenvalues $\lambda_{1}$ and $\lambda_{2}$,
with $\lambda_{1}>\lambda_{2}$. The aspect ratio of the ellipse is assumed to
be

$\displaystyle q~{}=~{}\sqrt{\frac{\lambda_{2}}{\lambda_{1}}}\,.$ (11)

The semi-major axis of the ellipse is assumed to be along the eigenvector
corresponding to $\lambda_{1}$. The square root is required here because the
eigenvalues of the inertia tensor tell us the squared distance along any
particular direction (see also equation 11 of Wu & Kroupa, 2013). There is
also an extra factor of the total number of particles, but this cancels upon
taking a ratio of eigenvalues.

Given the centre of the circular fit (Equation 8), the only remaining
parameter is its radius $r$. We estimate that

$\displaystyle r~{}=~{}\sqrt{\frac{S_{xx}+S_{yy}}{N}}\,.$ (12)

The semi-major axis $a$ of the ellipse is taken to be

$\displaystyle a~{}=~{}\frac{r}{\sqrt{q}}\,.$ (13)

This ensures that the initial guess circle and ellipse have the same area.

####  2.2.2 Finding the optimal parameters

For each point on the contour, we find the minimum distance $d_{i}$ to the
circle or ellipse by varying the azimuthal angle around it using a low-
resolution grid search, which we refine using gradient descent (Fletcher &
Powell, 1963). Once this has converged, we quantify the goodness of the fit
using

$\displaystyle S~{}\equiv~{}\sum_{i}{d_{i}}^{2}\,.$ (14)

The root mean square (rms) error of the circle or ellipse fit is thus
$\sqrt{S/N}$.

We optimize the ellipse fit using gradient descent in the five ellipse
parameters (centre position, semi-major axis, aspect ratio, and orientation).
The idea is the same for the best-fitting circle, though in this case we only
need to consider three parameters (centre position and radius). For this work,
the main parameters of interest are the aspect ratio $q_{\text{int}}$ of the
best-fitting ellipse and the extent to which this is an improvement over the
best-fitting circle.

##  3 Results

In the early stages of their evolution, both the isolated and EFE models
remain fairly circular. The models have a rotational period of 380 Myr at the
mass-weighted average of $r_{d}$ and $r_{g}$, so we investigated snapshots
after 3 rotational periods $-$ the non-axisymmetry mostly arises after this.
Most snapshots in our analysis have $q_{{}_{\text{int}}}\leq 0.90$. To
illustrate the possibility of a significantly elliptical contour, we only
present results where $q_{{}_{\text{int}}}\leq 0.86$ for the contour at the
level $\log_{10}\Sigma_{\text{gas}}=-0.6$ in units of
$M_{\odot}/\textrm{pc}^{2}$, which we chose as representative of the disc
outskirts. The contours at this level have a similar size to the isophote
plotted in figure 7 of Mancera Piña et al. (2022).

In the isolated model, spiral structure starts developing after $\approx 2$
Gyr, with clear spiral features are present in all snapshots after 3 Gyr (see
the left panels of Figure 1). Since AGC 114905 lacks clear spiral features, we
present snapshots from this model at 1300 Myr and 1800 Myr. We also show the
final snapshot at 5000 Myr, illustrating the spiral arms. These are generally
two-armed, but some snapshots contain three-armed spirals, especially around
the time when spiral structure is first apparent. This is in line with earlier
MOND simulations (Banik et al., 2020; Roshan et al., 2021).

Figure 1: Galaxy models in isolation (_left panels_) and including the EFE
with $g_{\textrm{ext}}=0.05\,a_{{}_{0}}$ at $30^{\circ}$ to the disc plane but
with no component along $y$ (_right panels_), shown face-on ($i=0^{\circ}$) at
the indicated times. In the model with the EFE, the coordinate system has been
rotated to view the galaxy along the direction of the total angular momentum
because the EFE causes the disc to precess (see section 4.2.5 of Banik et al.,
2020). The colours show the gas surface density as
$\log_{10}\Sigma_{\text{gas}}$, with $\Sigma_{\text{gas}}$ in units of
$M_{\odot}$/pc2. The solid red line shows the contour with level $-0.6$. The
dotted blue curve shows the best-fitting circle. The aspect ratio
$q_{\text{int}}$ of the best-fitting ellipse (solid black curve) is shown in
the legend as $\widetilde{i}\equiv\cos^{-1}q_{\text{int}}$, the inclination
that would be inferred by an observer seeing the galaxy face-on based only on
$q_{\text{int}}$ (Equation 15).

After accounting for the much more substantial barycentre drift and disc
precession (Section 2.1.1), the EFE model yields the snapshots shown in the
right panels of Figure 1. A second clump of mass appears after 2000 Myr and
then detaches from the galaxy by 3000 Myr $-$ the formation and detachment of
this clump are illustrated in Figure 2. However, the vast majority of the mass
remains in the main clump: it has a mass of $0.99\times 10^{9}\,M_{\odot}$ out
of a combined mass in both clumps of $1.36\times 10^{9}\,M_{\odot}$. The model
looks fairly regular both in the 1500 Myr snapshot before the second clump is
apparent and in the 4500 Myr snapshot after it has detached. We also show the
final snapshot at 5000 Myr for completeness.

Figure 2: Snapshots of our galaxy simulation with the EFE, showing the
formation of an overdense clump in the disc outskirts and its detachment to
form a secondary galaxy.

Despite viewing the galaxy face-on in both the isolated and EFE models, the
best-fitting elliptical contour has an aspect ratio indicative of a disc
inclined by $>32^{\circ}$, sometimes much more so (Table 2). In the isolated
model, the quality of the fit is however rather poor, as evidenced by the
almost identical goodness of fit using the best-fitting circle and ellipse. In
contrast, the disc in the EFE model has a much more regular appearance that is
fit much better with an ellipse than with a circle. This is particularly so
for the $\Sigma_{\text{gas}}$ contour in the 1500 Myr snapshot. Since the
best-fitting ellipse has a significantly flattened shape, assuming the galaxy
is axisymmetric would lead to a high inferred inclination of $59^{\circ}$.
Moreover, there are no clear spiral arms that would alert observers to the
possibility of a significantly overestimated inclination. This is also true
with the 4500 Myr and 5000 Myr snapshots, though weak spiral arms are evident
and the contour is not as well fit by an ellipse.

|  | rms error | rms error | Aspect |
---|---|---|---|---|---
| Snapshot | of circle | of ellipse | ratio |
Model | (Myr) | fit (kpc) | fit (kpc) | ($q_{\text{int}}$) | $\widetilde{i}$
Iso | 1300 | 3.10 | 3.15 | 0.87 | $33.9^{\circ}$
Iso | 1800 | 3.01 | 2.91 | 0.81 | $35.3^{\circ}$
Iso | 5000 | 6.49 | 4.27 | 0.47 | $61.9^{\circ}$
EFE | 1500 | 3.35 | 1.44 | 0.51 | $58.8^{\circ}$
EFE | 4500 | 2.04 | 1.57 | 0.68 | $46.5^{\circ}$
EFE | 5000 | 2.97 | 1.82 | 0.52 | $58.4^{\circ}$
Table 2: Parameters of the circle and ellipse fits to the contours shown on
Figure 1. The columns show the simulation snapshot used, the rms error of the
best-fitting circle and ellipse, and the aspect ratio of the latter. Assuming
an intrinsically circular contour would lead to the inferred inclination being
$\widetilde{i}$ (Equation 15), even though the observer actually sees the
galaxy face-on ($i=0^{\circ}$).

The significant deviation from axisymmetry in our models is consistent with
prior works: a similar deviation from axisymmetry can been in figure 8 of
Banik et al. (2020) and figure 1 of Wittenburg, Kroupa & Famaey (2020). Our
work differs in that the central surface density of our model is much lower in
order to match AGC 114905. However, the results remain qualitatively similar
because even LSB discs are still completely self-gravitating in MOND, allowing
them to sustain non-axisymmetric features like bars and spiral arms. Thus,
relying merely on the shape of the isophote to estimate $i$ could well cause a
significant overestimate.

##  4 Discussion

The sky-projected aspect ratio $q_{\text{sky}}$ of an isophote can be used to
estimate the inclination of a galaxy using

$\displaystyle\cos\widetilde{i}~{}\equiv~{}q_{\text{sky}}\,,$ (15)

where we use $\widetilde{i}$ to distinguish this estimate from the true $i$.
The $\widetilde{i}$ values recovered in this way for face-on Milgromian disc
galaxies are listed in Table 2. For a galaxy observed at low inclination from
slightly towards the minor axis, $q_{\text{sky}}=q_{\text{int}}\cos i$. Since
$q_{\text{int}}$ can easily be smaller than $\cos 32^{\circ}/\cos
11^{\circ}=0.86$, it is quite possible that a galaxy with
$\widetilde{i}=32^{\circ}$ actually has $i=11^{\circ}$. Our results indicate
that it could even be face-on.

This has significant implications for the RC estimated from line-of-sight
velocities because deriving the RC within the disc plane requires a geometric
correction factor of $1/\sin i$, which diverges at low $i$. The ‘observed’ RC
is based on a correction factor of $1/\sin\widetilde{i}$, which could be a lot
smaller as discussed above. The actual RC could thus be equal to the RC
reported by observers scaled up by a factor of $\sin\widetilde{i}/\sin i$,
which in the above example would be 2.8. For the case of AGC 114905, this
would raise the reported $v_{{}_{f}}=23$ km/s up to 64 km/s, quite close to
the MOND prediction that $v_{{}_{f}}=69$ km/s (Equation 4) for the estimated
$M_{b}=1.4\times 10^{9}M_{\odot}$ (Mancera Piña et al., 2022). The claimed
inconsistency with MOND arises because those authors obtain
$\widetilde{i}=32^{\circ}$ from an elliptical isophote, as illustrated in
their figure 7. Their argument is that the shape of the contour is such that
$\widetilde{i}$ cannot be as low as $11^{\circ}$. However, the actual $i$
could certainly be this low in a Milgromian universe because the isophotes of
a face-on galaxy may well be non-circular.

If as we argue AGC 114905 is nearly face-on, then even a small warp could
significantly alter the shape of its inferred RC if this is not accounted for.
Even rather isolated galaxies may become warped in MOND due to the EFE (Brada
& Milgrom, 2000; Banik et al., 2020). Therefore, very nearly face-on galaxies
should generally be excluded from RC studies. A cut of $i\geq 30^{\circ}$ was
recommended in Kroupa et al. (2018), but their recommendation is for galaxies
in the SPARC sample with much better observations that allow $i$ to be
estimated using a tilted ring fit to the 2D velocity field (Bosma, 1978). If
only the shapes of the isophotes are used, our work suggests that a better
criterion would be $i\geq 60^{\circ}$, which should provide some assurance
that the observed ellipticity of the image is indeed due to inclination,
especially if the isophotes can be fit well using ellipses with a fixed
centre, axis ratio, and major axis direction.

###  4.1 Overestimated inclinations in $\Lambda$CDM?

At face value, the observed RC published by Mancera Piña et al. (2022) is also
very problematic for $\Lambda$CDM, which like MOND prefers that $i=11^{\circ}$
(see their section 5.4). It is therefore interesting to consider whether AGC
114905 could be nearly face-on in a $\Lambda$CDM context. This would require
the apparent ellipticity to be a consequence of intrinsic non-axisymmetry in
the face-on view, which can be addressed using numerical simulations. One
problem is that if a galaxy is dominated by CDM, then its disc is essentially
not self-gravitating, making it difficult to sustain bars and spiral arms. The
fact that many LSB discs do have such features has been used to argue against
$\Lambda$CDM (see section 8 of McGaugh & de Blok, 1998).

Nowadays, it is possible to consider dwarf galaxies in a large cosmological
hydrodynamical simulation of the $\Lambda$CDM paradigm. Using zoom-in
simulations, Marasco et al. (2018) argued that triaxiality of the CDM halo
could cause a disc to be somewhat non-circular when viewed face-on, even if
the surface density is rather low. This is apparent in their figure 1, which
considers two galaxies with identifier numbers 19 and 17. By considering the
top row of this figure (galaxy 19), we found that the aspect ratio of the
plotted best-fitting ellipse to an isophote of $\Sigma_{\star}$ is $\approx
0.95$, which would not be sufficient to explain AGC 114905 because $\cos
32^{\circ}/\cos 11^{\circ}=0.86$. Moreover, we used the colours of the top
left panel of this figure to estimate that the central $\Sigma_{\star}\approx
55\,M_{\odot}/\textrm{pc}^{2}$, which is much higher than for AGC 114905 where
$\Sigma_{\star}\approx 7\,M_{\odot}/\textrm{pc}^{2}$ and
$\Sigma_{\textrm{gas}}\approx 3\,M_{\odot}/\textrm{pc}^{2}$ (see figure 1 of
Mancera Piña et al., 2022). We then considered the bottom row of figure 1 in
Marasco et al. (2018), which shows their galaxy 17. In this case, the aspect
ratio of the stellar component is $\approx 0.8$, which would in principle be
sufficient to reconcile the apparent ellipticity of AGC 114905 with an almost
face-on orientation. However, this galaxy has $\Sigma_{\star}\approx
80\,M_{\odot}/\textrm{pc}^{2}$, which is almost an order of magnitude larger
than the central baryonic surface density of AGC 114905. Since the above-
mentioned simulated galaxies in the study of Marasco et al. (2018) have a
similar amount of mass in gas as in the stars (see their section 3.1), their
baryonic surface density is much higher than for AGC 114905. Moreover, the
dissipative gas component may well be even closer to axisymmetric than the
stars. In summary, the results of Marasco et al. (2018) apply to ‘typical’
dwarf irregulars and thus cannot be extrapolated to the case of AGC 114905,
which has a much lower stellar surface density and a larger disc than the
simulated systems they studied. Further work is needed to clarify whether
sufficiently non-circular gas discs with an extremely low surface density can
spontaneously form in the $\Lambda$CDM framework.

Recently, this issue was investigated by Sellwood & Sanders (2022) using
Newtonian $N$-body simulations tailored to the properties of AGC 114905. In
their model where $i=15^{\circ}$ and the CDM halo is tuned to reproduce the
resulting higher RC, the galaxy appears almost completely axisymmetric.
However, $i=15^{\circ}$ is already rather high: Mancera Piña et al. (2022)
argue in their section 5.4 that AGC 114905 follows the expected relation
between stellar and halo mass only if $i\approx 11^{\circ}$. Essentially, a
low inclination is needed to reconcile the high expected RC amplitude with the
small observed line-of-sight velocity gradient across AGC 114905. Using a
lower inclination than in Sellwood & Sanders (2022) would however make the
disc even less relevant to the total potential, making it harder to understand
how the isophotes can deviate from axisymmetry at the 14% level. Thus, the
preliminary results of Sellwood & Sanders (2022) suggest that it is very
difficult to generate a significant amount of non-axisymmetry in a galaxy like
AGC 114905 where the CDM needs to dominate because
$\Sigma_{\star}+\Sigma_{\textrm{gas}}\ll\Sigma_{M}$ (Equation 5). It is not
yet clear whether considering a somewhat triaxial CDM halo could yield the
required departure from axisymmetry if the degree of triaxiality and the
orientation of the disc relative to the halo follow expectations in a broader
cosmological context.

##  5 Conclusions

We conduct hydrodynamical MOND simulations of a gas-rich disc galaxy with a
similar mass distribution to AGC 114905, first in isolation and then including
the EFE with $g_{\textrm{ext}}=0.05\,a_{{}_{0}}$ directed at $30^{\circ}$ to
the disc plane. We show that isodensity contours in $\Sigma_{\text{gas}}$ can
appear to be elliptical at the 25% level or more in the face-on view where
$i=0^{\circ}$. This could mislead observers into thinking that $i$ is as high
as $60^{\circ}$. As RCs require a correction factor of $1/\sin i$, this could
cause the true RC to greatly exceed that reported in the observational
literature. In the specific case of AGC 114905, this invalidates the claim of
Mancera Piña et al. (2022) that the low amplitude of its RC falsifies MOND
because this claim relies on $i=32^{\circ}$. The RC would be much higher and
consistent with MOND if instead $i=11^{\circ}$, which as shown in this work is
quite possible. This is due to the fact that in MOND, even LSB discs are self-
gravitating, allowing them to sustain non-axisymmetric features like bars and
spiral arms (Banik et al., 2020). Observed LSB discs often do have such
features (McGaugh, Schombert & Bothun, 1995). A related phenomenon in our
simulations is that satellites can split off from the outskirts of a gas-rich
disc galaxy in a process similar to the gravitational instability model of gas
giant planet formation (Figure 2). Such satellites should be orbiting close to
the disc plane. It is intriguing that a galaxy not subject to tidal forces can
nevertheless develop features resembling tidal tails.

Given the hundreds of RCs that agree with MOND (e.g., Li et al., 2018) and the
inevitable outliers from the correct theory given imperfect data (e.g., Kroupa
et al., 2018; Cameron et al., 2020), MOND is not presently in significant
tension with galaxy RCs. We conclude that nearly face-on galaxies are
unsuitable for such tests, especially in a theory where significant departures
from axisymmetry are quite possible because galaxies are completely self-
gravitating at any surface brightness.

## Acknowledgements

IB is supported by Science and Technology Facilities Council grant
ST/V000861/1, which also partially supports HZ. The authors thank Ingo Thies,
Moritz Haslbauer, Elena Asencio, and Jerry Sellwood for useful discussions.
They are also grateful for comments from the anonymous referee which helped to
substantially improve this article.

## Data availability

The algorithms used to prepare and run por simulations of disc galaxies and to
extract their results into human-readable form are publicly
available.222https://bitbucket.org/SrikanthTN/bonnpor/src/master/ A user
manual is available describing the operation of these codes (Nagesh et al.,
2021).

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