Construction of blow-up solution for 5 dimentional critical fujita type equation with different blow-up speed

Liqun Zhang Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, AMSS, and School of Mathematical Sciences, UCAS, Beijing 100190, China 123 lqzhang@math.ac.cn and Jianfeng Zhao Institute of Mathematics, AMSS, UCAS, Beijing 100190, China 123 zhaojianfeng@amss.ac.cn

Abstract.

We are concerned with blow-up solutions of the 5-dimensional energy critical heat equation $u_{t}=\Delta u+|u|^{\frac{4}{3}}u$ . Our main result is to show that the existence of type II solutions blows up at 2 points with 2 different blow-up rates. The inner-outer gluing method has been employed.

1. Introduction

We consider the equation

(1.1)

u_{t}=\Delta u+|u|^{p-1}u\ \ \ \ in\in\Omega\times(0,T).

Early in 1966, Fujita [11] started the research about equation (1.1). If we only consider the time variable, (1.1) becomes an ordinary differential equations $\dot{g}=|g|^{p-1}g$ . It’s solution

g(t)=(p-1)^{-\frac{1}{p-1}}(T-t)^{-\frac{1}{p-1}}

blows up at time T, which is so called the type I blow-up. And if a blow-up solution does not satisfies this blow-up rate is called type II blow-up. It’s known type II blow-up is faster than type I blow-up. The more result about type I blow-up can be seen in [14, 13, 22, 29, 25].

Let $p_{S}=\frac{n+2}{n-2}$ denote the critical Sobolev exponent. In subcritical case, Giga and Kohn [12] first prove in convex domain, only type I blow-up occurs. In super critical case, let $p_{JL}$ denote the Joseph-Lundgreen exponent

p_{JL}=\begin{cases}\infty,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3\leq n\leq 10,\\ 1+\frac{4}{n-4-2\sqrt{n-1}},\ \ \ \ \ \ \ \ \ \ n\geq 11.\end{cases}

The first example of type II solution is given by M.A. Herrero and Velázquez [19, 20], where they constructed a positive radial solution in the case $p>p_{JL}$ . Collot [1] constructed a type II solution with the same profile later. In the case $p_{S}<p<p_{JL}$ , Matano and Merle [21] excluded type II blow-up in the radial case. For the case $p=p_{JL}$ , Seki constructed a blow-up solution in [27]. For the case $p=p_{S}$ , Filippas, Herrero and Velázquez [10] proved that the radial positive solution can only be a type I blow-up solution. By using the asymptotic matching method, they obtained formally the possible blow-up solution with the blow-up rates given by (after a slightly modify):

(1.2)

\|u(\cdot,t)\|_{\infty}=\begin{cases}(T-t)^{-k},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n=3,\\ (T-t)^{-k}|\ln(T-t)|^{\frac{2k}{2k-1}},\ \ \ \ \ \ n=4,\\ (T-t)^{-3k},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n=5,\\ (T-t)^{\frac{5}{2}}|\ln(T-t)|^{\frac{15}{4}},\ \ \ \ \ \ \ \ \ \ n=6,\end{cases}

where $k=1,2,\cdots$ . When $n\geq 7$ , Collot, Merle and Raphaël [2] proved the type II blow-up solution can’t be around the ground state. For n=4 type II solutions has been constructed by Schweyer [26] in 2012, In the case $3\leq n\leq 6$ , type II solutions has been constructed by Del Pino, Musso, Wei and Zhou [9], they have construct a multi point blow-up in 2019. The case n=5 is first construct by Del Pino, Musso, Wei [6] in 2018, with the blow up rate of k=1 in (1.2). Junichi Harada [17] finished the construction of higher blow-up rate for n=5 in 2019. And Junichi Harada [18] also proved the case of n=6. Very recently Del Pino, Musso, Wei, Zhang and Zhou [8] finished the proof of n=3.

In the above mentioned construction of the critical case, except the construction of Schweyer, they all employed the so called inner-outer gluing method. This technique is not only applied to the construction of finite time blow-up solution for the critical case, but also to super critical case and infinity time blow-up [3, 7]. Beside Fujita equation, Dávila, Del Pino and Wei also construct a blow-up solution for harmonic map equation [4].

1.1. Main result

For 5 dimensional energy critical equation, our main result is

Theorem 1.1.

Let $\Omega=\mathbb{R}^{n}$ , n = 5, $p=\frac{7}{3}$ be the critical exponent. For any given two points 0, q in $\mathbb{R}^{5}$ , then for any sufficiently small $T>0$ , there exists an initial data $u_{0}$ , such that the solution of equation (1.1) blows up at time T, where the blow-up points are at the given 0 and q with different blow-up rate. Moreover, the main order of the solution is as

u(x,t)=\sum_{j=1,2}U_{\lambda_{j}(t),\xi_{j}(t)}(x)-Z_{1}\eta_{1}-Z_{2}+\theta(x,t),

when $t\rightarrow T$

\lambda_{j}(t)\rightarrow 0,\ \ \ \ \xi_{1}(t)\rightarrow 0,\ \ \ \ \xi_{2}(t)\rightarrow q,

and $\lambda_{1}(t)\sim C_{1}(T-t)^{4}$ , $\lambda_{2}(t)\sim C_{2}(T-t)^{2}$ , with

\theta(x,t)\in L^{\infty}(\mathbb{R}^{5}\times(0,T)),

The definition of $Z_{1}$ , $Z_{2}$ and $\eta_{1}$ are left to section 2.

Remark 1.2.

In our construction, actually we can construction any k points blow-up with one point has blow-up rate $(T-t)^{-6}$ and the others have blow-up rate $(T-t)^{-3}$ .

2. Basic facts in the construction

As we all known, in critical case, the steady solution of equation (1.1) is unique, up to scalings and translations.

(2.1)

U(x)=\alpha_{n}({\frac{1}{1+|x|^{2}}})^{\frac{n-2}{2}},\alpha_{n}=(n(n-2))^{\frac{n-2}{4}},

which is called Talenti-Aubin steady state [28].

2.1. The Construction of the approximated solutions

Fixed a point $q\in\mathbb{R}^{5}$ , we hope to find a solution of this form:

(2.2)

u(x,t)=\sum_{j=1,2}U_{\lambda_{j}(t),\xi_{j}(t)}-Z_{1}\eta_{1}-Z_{2}+\theta(x,t),

where

U_{\lambda_{j}(t),\xi_{j}(t)}={\lambda_{j}(t)}^{-\frac{n-2}{2}}U(\frac{x-\xi_{j}(t)}{\lambda_{j}(t)}).

$\lambda_{j}(t)$ are the scaling parameters, $\xi_{j}(t)$ are the translation parameters, translation is small compare with scaling, that is, $\xi_{j}(t)=o(\lambda_{j}(t))$ .
$Z_{1}:=M(T-t)e_{1}$ , where $M$ is a small constant, $e_{1}=1-\frac{|z|^{2}}{10}$ , $z=\frac{x}{\sqrt{T-t}}$ is the self similar variable. Here $Z_{1}$ is actually a solution of heat equation. Since the solution does not decay at the infinity, we need to cut it off by multiplying the cut off function $\eta_{1}=\eta(\frac{x}{(T-t)^{\frac{1}{8}}})$ , where $\eta(x)=1$ if $x\leq 1$ , and $\eta(x)=0$ if $x\geq 2$ and it’s smooth in the interval $(1,2)$ .
We let $Z_{2}$ be the unique solution of

(2.3)

\begin{cases}\partial_{t}Z_{2}=\Delta Z_{2}\ \ \ \ (x,t)\in\mathbb{R}^{5}\times(0,\infty),\\ Z_{2}(\cdot,0)=Z_{2,0}\ \ \ \ x\in\mathbb{R}^{5}.\end{cases}

where $Z_{2,0}$ is an odd function with compact support. $Z_{2,0}$ has a small $L^{\infty}$ norm. It satisfies $Z_{2,0}(q)>\frac{\|Z_{2}\|_{\infty}}{2}$ in the interval [0,T]. We can obtain it by use the continue property of heat equation. By uniqueness, it is easy to see $Z_{2}$ is an odd function.

2.2. Inner-outer gluing

We write the remainder in the form

(2.4)

\theta(x,t)=\lambda_{1}^{-\frac{n-2}{2}}\phi_{1}(y_{1},t)\eta_{R}(y_{1})+\lambda_{2}^{-\frac{n-2}{2}}\phi_{2}(y_{1},t)\eta_{R}(y_{1})+\psi(x,t),

where $y_{1}=\frac{x-\xi_{1}}{\lambda_{1}}$ . We denote $\eta_{R}(y_{1})=\eta_{R,1}$ , and also $\eta_{R}(y_{2})=\eta_{R,2}$ later.
We require $\psi$ be a continuous function and has the following estimate

(2.5)

\begin{split}|\psi|\leq\begin{cases}\delta_{0}(T-t)(1+|z|^{4})\ &\ |z|\leq(T-t)^{-\frac{1}{4}},\\ \frac{\delta_{0}}{1+|x|^{2}}\ &\ |z|>(T-t)^{-\frac{1}{4}},\end{cases}\end{split}

where

\delta_{0}=\frac{\|Z_{2}\|_{\infty}}{10}.

Let us set the error function

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

Denote

y_{i}=\frac{x-\xi_{i}}{\lambda_{i}},

(2.6)

E_{i}(y_{i},t)=\lambda_{i}\dot{\lambda}_{i}[(\frac{3}{2}U(y_{i})+y_{i}\cdot\nabla U(y_{i}))]+\lambda_{i}\dot{\xi}_{i}\cdot\nabla U(y_{i}).

We compute $S(u)$

\begin{split}&S(\sum_{i=1,2}U_{\lambda_{i}(t),\xi_{i}(t)}-Z_{1}\eta_{1}-Z_{2}+\theta(x,t))\\ &=-\theta_{t}+\Delta\theta+\sum_{i=1,2}pU_{\lambda_{i},\xi_{i}}^{p-1}(\theta(x,t)-Z_{1}\eta_{1}-Z_{2})+\sum_{i=1,2}\lambda_{i}^{-\frac{7}{2}}E_{i}+N\\ &=\sum_{i=1,2}\eta_{R,i}\lambda_{i}^{-\frac{7}{2}}(-\lambda_{i}^{2}\partial_{t}\phi_{i}+\Delta_{y}\phi_{i}+pU^{p-1}[\phi_{i}+\lambda_{i}^{\frac{3}{2}}(-Z_{1}\eta_{1}-Z_{2}+\psi)]+E_{i})\\ &-\psi_{t}+\Delta_{x}\psi+\sum_{i=1,2}p\lambda_{i}^{-2}(1-\eta_{R,i})U^{p-1}(-Z_{1}\eta_{1}-Z_{2}+\psi)+\sum_{i=1,2}A[\phi_{i}]+\sum_{i=1,2}B[\phi_{i}]\\ &+\sum_{i=1,2}\lambda_{i}^{-\frac{7}{2}}E_{i}(1-\eta_{R,i})+N.\end{split}

Where

A[\phi_{i}]:=\lambda_{i}^{-\frac{7}{2}}[\Delta_{y_{i}}\eta_{R,i}\phi_{i}+2\nabla_{y_{i}}\eta_{R,i}\nabla_{y_{i}}\phi_{i}],

\begin{split}B[\phi_{i}]:=&\lambda_{i}^{-\frac{5}{2}}[\dot{\lambda}_{i}({y_{i}}\cdot\nabla_{y_{i}}\phi_{i}+\frac{3}{2}\phi_{i})\eta_{R,i}+\dot{\xi}_{i}\cdot\nabla_{y_{i}}\phi_{i}\eta_{R,i}+(\dot{\lambda}_{i}{y_{i}}\cdot\nabla_{y_{i}}\eta_{R,i}+\dot{\xi}_{i}\cdot\nabla_{y_{i}}\eta_{R,i})\phi_{i}],\\ N:=&|U_{\lambda_{1},\xi_{1}}+U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2}|^{p-1}(U_{\lambda_{1},\xi_{1}}+U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}\\ &-Z_{2})-U_{\lambda_{1},\xi_{1}}^{p}-U_{\lambda_{2},\xi_{2}}^{p}-pU_{\lambda_{1},\xi_{1}}^{p-1}(\theta-Z_{1}\eta_{1}-Z_{2})-pU_{\lambda_{2},\xi_{2}}^{p-1}(\theta-Z_{1}\eta_{1}-Z_{2})\\ &+\frac{\partial\eta_{1}}{\partial t}Z_{1}+\nabla_{x}Z_{1}\nabla\eta_{1}-\Delta\eta_{1}Z_{1}.\end{split}

We shall find a solution of (1.1) if we find a pair $(\phi_{1}(y,t),\phi_{2}(y,t),\psi(x,t))$ solves the following system of equations

(2.7)

\begin{cases}\lambda_{1}^{2}\partial_{t}\phi_{1}=\Delta_{y_{1}}\phi_{1}+pU^{p-1}(y_{1})+H_{1}(\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2})\ \ \ (y_{1},t)\in\ B_{2R}(0)\times(0,T),\\ \phi_{1}(y,0)=\phi_{1,0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_{1}\in B_{2R}.\end{cases}

(2.8)

\begin{cases}\lambda_{2}^{2}\partial_{t}\phi_{2}=\Delta_{y_{2}}\phi_{2}+pU^{p-1}(y_{2})+H_{2}(\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2})\ \ \ (y_{2},t)\in\ B_{2R}(0)\times(0,T),\\ \phi_{2}(y,0)=\phi_{2,0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_{2}\in B_{2R}.\end{cases}

(2.9)

\begin{cases}\psi_{t}=\Delta_{x}\psi+G(\phi,\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2})\quad(x,t)\in\mathbb{R}^{5}\times(0,T),\\ \psi(\cdot,0)=\psi_{0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in\mathbb{R}^{5}.\end{cases}

Where for $i=1,2$

(2.10)

H_{i}(\psi,\lambda_{i},\xi_{i}):=\lambda_{i}^{\frac{3}{2}}pU(y_{i})^{p-1}(-Z_{1}(\xi_{i}+\lambda_{i}y_{i})-Z_{2}(\xi_{i}+\lambda_{i}y_{i})+\psi(\xi_{i}+\lambda_{i}y_{i}))+E_{i}(y_{i},t),

and

(2.11)

\begin{split}G(\phi_{1},\phi_{2},\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2}):=&\sum_{i=1,2}p\lambda_{i}^{-2}(1-\eta_{R,i})U(y_{i})^{p-1}(-Z_{1}\eta_{1}-Z_{2}+\psi)\\ &+\sum_{i=1,2}A[\phi_{i}]+\sum_{i=1,2}B[\phi_{i}]+\sum_{i=1,2}\lambda_{i}^{-\frac{7}{2}}E_{i}(1-\eta_{R,i})+N.\end{split}

2.3. A result about the linearized equation

To deal with the inner problemma, we consider the corresponding linear problem of (1.1).

(2.12)

\begin{split}\begin{cases}\lambda^{2}\phi_{t}&=\Delta_{y}\phi+pU(y)^{p-1}\phi+h(y,t)\ \ \ \ (y,t)\in\ B_{2R}\times[0,T),\\ \phi(y,0)&=lW_{0}(y)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y\in B_{2R},\end{cases}\end{split}

where $l$ is a constant be choosing later, $W_{0}$ is the eigenfunction of $L_{0}=\Delta+pU^{p-1}$ with the negative eigenvalue.
It is known that

W_{0}(y)\sim|y|^{-2}e^{-\sqrt{\lambda_{0}}|y|},\ \ \ \ as\ y\rightarrow\infty,

where $\lambda_{0}$ is the only negative eigenvalue of $L_{0}$ .
And $h(y,t)$ satisfies the orthogonal condition

(2.13)

\int_{B_{2R}}h(y,t)W_{i}(y)dy=0\ \ \ \ i=1,\cdots,n+1,\ \ \ \ t\in\ [0,T).

We define

\|h\|_{a,\nu}:=\sup_{y\in B_{2}R,t\in[0,T)}\lambda_{0}^{\nu}(1+|y|^{a})|h(y,t)|,

and

\|\phi\|_{*a,\nu}:=\sup_{y\in B_{2}R,t\in[0,T)}\lambda_{0}^{\nu}\frac{(1+|y|^{6})}{R^{6-a}}|\phi(y,t)|.

Actually we expect solution $\phi$ in the space where the $\|\cdot\|_{a,\nu}$ is finite. However, we can only know the estimate due to Del Pino, Musso and Wei, there they proved $\phi$ is in $\|\cdot\|_{*a,\nu}$ . We copy the lemma of Del Pino, Musso and Wei here

Lemma 2.1.

There exists constant $C_{3}>0$ , for all sufficient large $R>0$ , if $\|h\|_{2+a,\nu}<+\infty$ , and h satisfies orthogonal condition, then there exists linear operator L satisfies

\phi=L^{in}[h],\ \ \ \ l=l[h],

fulfill the equation, and l[h] satisfies

|l[h]|+\|(1+y)\nabla_{y}\phi\|_{*a,\nu}+\|\phi\|_{*a,\nu}\leq C_{3}\|h\|_{2+a,\nu}.

2.4. Local properties of the heat equation

In order to solve the outer problem, we shall first list some known results for the completeness of the paper. We will not restrict in dimension $n=5$ . Consider the heat equation

\begin{split}u_{t}=\Delta u\ \ \ \ (x,t)\in\mathbb{R}^{n}\times(0,\infty).\end{split}

We make use of self-similar variables

(2.14)

w(z,\tau)=u(x,t),\ \ \ z=\frac{x}{\sqrt{T-t}},\ \ \ T-t=e^{-\tau}.

The function $w(z,\tau)$ solves

\begin{split}w_{\tau}=A_{z}w\ \ \ \ (z,\tau)\in\mathbb{R}^{n}\times(0,\infty),\end{split}

where $A_{z}=\Delta_{z}-\frac{z}{2}\cdot\nabla$ . We define the weighted $L^{2}$ space

L_{\rho}^{2}(\mathbb{R}^{n}):=\{f\in L_{loc}^{2}(R^{n}),\|f\|_{\rho}<\infty\},\ \|f\|_{\rho}^{2}=\int_{R^{n}}f^{2}(z)\rho(z)dz,\

where $\rho(z)=e^{-\frac{|z|^{2}}{4}}$ . And inner product is denoted by

(f_{1},f_{2})_{\rho}:=\int_{R^{n}}f_{1}(z)f_{2}(z)\rho(z)dz.

Consider the eigenvalue problem of $A_{z}$ in $L_{\rho}^{2}(\mathbb{R}^{n})$ . Let $e_{\alpha}$ be the eigenfunction

-A_{z}e_{\alpha}=\lambda_{\alpha}e_{\alpha},

where $\alpha$ is multi-index, $\alpha=(\alpha_{1},\cdots,\alpha_{n})\in\ \mathbb{Z}^{n}$ , corresponding eigenvalue is

(2.15)

\lambda_{\alpha}=\frac{|\alpha|}{2},\ \ \ \ |\alpha|=\alpha_{1}+\cdots+\alpha_{n}.

The eigenfunction

e_{\alpha}=\prod e_{\alpha_{i}},\ \ \ \ e_{\alpha_{i}}=H_{\alpha_{i}},

where $H_{\alpha_{i}}$ is the $\alpha_{i}$ -th Hermite polynomial. Then $(T-t)^{\lambda_{\alpha}}e_{\alpha}$ is a solution of heat equation. We denote $e^{A_{z}}f_{0}$ to be the solution with initial data $f_{0}$ , by Duhamel’s principle

(2.16)

e^{A_{z}}f_{0}=\frac{c_{n}}{(T-e^{-\tau})^{\frac{n}{2}}}\int_{R^{n}}e^{-\frac{|e^{-\frac{\tau}{2}}z-\zeta|^{2}}{4(T-e^{-\tau})}}f_{0}(\zeta)d\zeta.

By using this formula, It is easy to obtain the following estimate.

Lemma 2.2.

There exists a constant $C=C_{n}>0$ , such that

(2.17)

|e^{A_{z}(\tau-\tau_{0})}f_{0}|\leq C_{n}\frac{e^{\frac{e^{-(\tau-\tau_{0})|z|^{2}}}{4(2-T+e^{-\tau})}}}{(T-e^{-\tau})^{\frac{n}{4}}}\|f_{0}\|_{\rho}.\ \ \ \ \ \ \ (z,\tau)\in\mathbb{R}^{n}\times(0,\infty)

The proof of this lemma can be seen in [16].

Lemma 2.3.

For any $l\in\mathbb{Z}^{+}$ , it has

(2.18)

|e^{A_{z}(\tau-\tau_{0})}|z|^{2l}|\leq C_{l}(1+e^{-l(\tau-\tau_{0})}|z|^{2l}).\ \ \ \ \ \ \ (z,\tau)\in\mathbb{R}^{n}\times(0,\infty)

The proof of this lemma can be seen in [15].
The last one is an estimate of heat equation in different scaling variable.

Lemma 2.4.

Consider nonhomogeneous heat equation

(2.19)

\begin{cases}\partial_{t}\Phi-\Delta\Phi=g\ \ \ \ \ (x,t)\in\mathbb{R}^{n}\times(0,T),\\ \Phi\mid_{t=0}=0\ \ \ \ \ \ \ \ \ \ \ x\in\mathbb{R}^{n}.\end{cases}

Suppose

|g|\leq\frac{1}{\lambda^{2}(1+|y|^{2+a})},

then there exist constants $C_{1}$ , ${\gamma}>0$ such that

|\Phi|\leq C_{1}(T^{\gamma}+\frac{1}{1+|y|^{a}}).

The proof of this lemma can be seen the lemma 4.2 in [6].

3. The parameters $\lambda_{i}$ and $\xi_{i}$

In this section we shall solve the scaling parameter $\lambda_{i}$ and translation parameter $\xi_{i}$ . They essentially determine the blow up rates.

3.1. Formal derivation of $\lambda_{i}$ and $\xi_{i}$

We formally drive approximate equation of $\lambda_{i}$ and $\xi_{i}$ , and determine the first order of the parameter.

Recall the inner problem

\begin{split}\lambda_{i}^{2}\partial_{t}\phi_{i}=\Delta_{y_{i}}\phi_{i}+pU^{p-1}(y_{i})+H_{i}(\psi,\lambda_{i},\xi_{i})\ \ \ (y_{i},t)\in B_{2R}\times(0,T),\end{split}

and the notation

y_{i}=\frac{x-\xi_{i}}{\lambda_{i}},

where $i=1,2$ and

H_{i}(\psi,\lambda,\xi):=\lambda_{i}^{\frac{3}{2}}pU(y_{i})^{p-1}(-Z_{1}(\xi_{i}+\lambda_{i}y_{i})-Z_{2}(\xi_{i}+\lambda_{i}y_{i})+\psi(\xi_{i}+\lambda y_{i}))+E_{i}(y_{i},t).

Because it is a type II blow-up, then

|\lambda_{i}|<<\sqrt{T-t}.

Therefore it is reasonable for us to ignore $\lambda_{i}^{2}\partial_{t}\phi_{i}$ , the equation becomes an elliptic equation

(3.1)

\Delta_{y_{i}}\phi_{i}+pU^{p-1}(y_{i})+H_{i}(\psi,\lambda_{i},\xi_{i})=0.

From Fredholm theorem, the solvability condition of this equation is the nonhomogeneous term orthogonal to the kernel of the operator.
We let

L_{0}u=\Delta u+pU^{p-1}u.

The kernels of $L_{0}$ are

(3.2)

W_{j}(x)=\frac{\partial U(x)}{\partial x}\ \ \ \ j=1,2,3,4,5,

and

(3.3)

W_{6}(x)=\frac{\partial U_{\lambda}(x)}{\partial\lambda}\mid_{\lambda=1}=(\frac{3}{2}U(x)+x\cdot\nabla U(x)).

The orthogonal conditions are approximate as

(3.4)

\int_{R^{5}}H_{i}(y_{i},t)W_{j}(y_{i})dy=0\ \ \ \ j=1,\ldots,6,\quad i=1,2.

Because $Z_{2}$ is an odd function, then $Z_{2}(0,t)=0$ . In $y_{1}\in B_{2R}$ , by the mean value theorem, $|Z_{2}(\lambda_{1}y_{1},t)|\leq 2R|\lambda_{1}|$ is a higher order term which could be omitted. We require remainder $\psi$ can not make a big influence and also be omitted. Notice that

Z_{1}=M(T-t)(1-\frac{|z|^{2}}{10}),

in the region $B_{2R}$ it is approximate as $M(T-t)$ .
And from (2.5)

E_{i}(y_{i},t)=\lambda_{i}\dot{\lambda}_{i}[(\frac{3}{2}U(y_{i})+y_{i}\cdot\nabla U(y_{i}))]+\lambda_{i}\dot{\xi}_{i}\cdot\nabla U(y_{i}).

Also

W_{6}(y_{i})=(\frac{3}{2}U(y_{i})+y_{i}\cdot\nabla U(y_{i}))

is an even function, while $W_{j}(y_{i})=\frac{\partial U(y_{i})}{\partial y_{i}}$ is odd in variable $y_{i}$ .
So

\int_{R^{5}}H_{1}(y_{1},t)W_{6}(y_{1})dy=0

can approximately be written as

\dot{\lambda}_{1}\lambda_{1}\int_{R^{5}}W_{6}^{2}-M(T-t)\lambda_{1}^{\frac{3}{2}}\int_{R^{5}}pU(y_{1})^{p-1}W_{6}(y_{1})dy_{1}=0.

Let

M\frac{\int_{R^{5}}pU(y_{1})^{p-1}W_{6}(y_{1})dy_{1}}{\int_{R^{5}}W_{6}^{2}}=-\frac{3M}{2}\frac{\int_{R^{5}}U(y_{1})^{p}dy_{1}}{\int_{R^{5}}W_{6}^{2}}:=-\kappa_{1}.

Then the main order of $\lambda_{1}$ is

\lambda_{1,0}=\frac{\kappa_{1}^{2}(T-t)^{4}}{16}.

Next we deal with $\lambda_{2}$ , because $Z_{1}\eta_{1}$ is supported around 0, but the integration region is around q, so we can omit $Z_{1}\eta_{1}$ . And we also omit the remainder $\psi$ . $Z_{2}$ can be approximated as $Z_{2,0}(q)$ around $q$ , then

\int_{R^{5}}H_{2}(y_{2},t)W_{6}(y_{2})dy=0

can be approximated by

\dot{\lambda}_{2}\lambda_{2}\int_{R^{5}}W_{6}^{2}-\lambda_{2}^{\frac{3}{2}}\int_{R^{5}}pU(y_{2})^{p-1}W_{6}(y_{2})Z_{2,0}(q)dy_{2}=0.

Let

\frac{\int_{R^{5}}pU(y_{2})^{p-1}W_{6}(y_{2})Z_{2,0}(q)dy_{2}}{\int_{R^{5}}W_{6}^{2}}=-Z_{2,0}(q)\frac{\int_{R^{5}}U(y_{2})^{p}dy_{2}}{\int_{R^{5}}W_{6}^{2}}=-\kappa_{2}.

Then the main order of $\lambda_{2}$ is

\lambda_{2}=\frac{\kappa_{2}^{2}(T-t)^{2}}{4}.

Similarly about

\int_{R^{5}}H_{i}(y_{i},t)W_{j}(y_{2})dy=0,\quad i=1,2,\quad j=1,2,3,4,5,

we obtain

|\xi_{1}|=O(\lambda_{1}^{\frac{3}{2}}),

and

|\xi_{2}|=q+O(\lambda_{2}^{\frac{3}{2}}).

3.2. The selection of $\lambda_{i}$ and $\xi_{i}$

Recall the orthogonal condition

\begin{split}\int_{B_{2R}}H_{i}Z_{j}=0\ \ \ \ i=1,2\ \ \ \ j=1,2,3,4,5,6,\\ \end{split}

For given $\psi$ , define $\lambda_{i}$ , $\xi_{i}$ as the unique solution of the above orthogonal condition. We require

(3.5)

\lambda_{i}(T)=0,\ \ \ \ \xi_{1}(T)=0,\ \ \ \ \xi_{2}(T)=q.

Divide $\lambda_{i}$ into

\lambda_{i}=\lambda_{i,0}+\lambda_{i,1}.

For $\lambda_{1}$

(3.6)

\begin{split}(\dot{\lambda}_{1,0}+\dot{\lambda}_{1,1})\int_{B_{2R}}W_{6}^{2}-\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})(Z_{1}+Z_{2}-\psi)dy_{1}=0.\end{split}

Then by the choosing of $\lambda_{1,0}$ , we have

\dot{\lambda}_{1,0}+\kappa_{1}(T-t)\lambda_{1,0}^{\frac{1}{2}}=0.

Substitute it into equation (3.6)

(3.7)

\begin{split}&\dot{\lambda}_{1,1}\int_{R^{5}}W_{6}^{2}-\lambda_{1}^{\frac{1}{2}}\int_{R^{5}}pU^{p-1}W_{6}M(T-t)dy_{1}\\ &+\lambda_{1,0}^{\frac{1}{2}}\int_{R^{5}}pU^{p-1}W_{6}M(T-t)dy_{1}-\int_{R^{5}/B_{2R}}Z_{6}^{2}(\dot{\lambda}_{1,0}+\dot{\lambda}_{1,1})dy_{1}\\ &-\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})[M(T-t)(-\frac{|z|^{2}}{10})+Z_{2}-\psi]dy_{1}=0.\end{split}

That is

(3.8)

\begin{split}&\dot{\lambda}_{1,1}\int_{R^{5}}W_{6}^{2}+\frac{M(T-t)\lambda_{1,1}}{(\lambda_{1}^{\frac{1}{2}}+\lambda_{1,0}^{\frac{1}{2}})}\int_{R^{5}}pU^{p-1}W_{6}dy_{1}-\int_{R^{5}/B_{2R}}W_{6}^{2}(\dot{\lambda}_{1,0}+\dot{\lambda}_{1,1})dy_{1}\\ &-\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})[M(T-t)(-\frac{|z|^{2}}{10})+Z_{2}-\psi]dy_{1}=0.\end{split}

Because in $y_{1}\in B_{2R}$ , $|z|\leq 2R(T-t)^{\frac{7}{2}}$ , then

(3.9)

\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})[M(T-t)(-\frac{|z|^{2}}{10})]\leq\lambda_{1}^{\frac{1}{2}}(T-t)^{8}.

Since $Z_{2}$ is odd, thus

\begin{split}&\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})Z_{2}(x+\xi_{1})dy_{1}=\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})Z_{2}(y_{1}\lambda_{1})dy_{1}=0.\end{split}

Therefore

(3.10)

\begin{split}&|\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})Z_{2}dy_{1}|\\ &=|\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})[Z_{2}(x)-Z_{2}(y_{1}\lambda_{1})]dy_{1}|\\ &\lesssim|\xi_{1}\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})\nabla Z_{2}(s)dy_{1}|\\ &\lesssim|\xi_{1}\lambda_{1}^{\frac{1}{2}}\int_{B_{2R}}pU(y_{1})^{p-1}W_{6}(y_{1})dy_{1}|.\end{split}

By (3.9)-(3.10) and the definition of $\psi$ in (2.5), equation (3.8) can be written as

(3.11)

\begin{split}&\dot{\lambda}_{1,1}\int_{R^{5}}W_{6}^{2}+\int_{R^{5}}U^{p}dy_{1}M(T-t)\frac{\lambda_{1,1}}{2\lambda_{1,0}^{\frac{1}{2}}}+\delta_{R,T,\delta_{0}}(\dot{\lambda}_{1,0}+\dot{\lambda}_{1,1}+(T-t)\lambda_{1}^{\frac{1}{2}})\\ &-\int_{R^{5}}U(y_{1})^{p}dy_{1}M(T-t)\frac{\lambda_{1,1}^{2}}{2\lambda_{1,0}^{\frac{1}{2}}(\lambda_{1}^{\frac{1}{2}}+\lambda_{1,0}^{\frac{1}{2}})^{2}}=0,\end{split}

where when $R\rightarrow\infty$ , $T\rightarrow 0$ , $\delta_{0}\rightarrow 0$ , $\delta_{R,T,\delta_{0}}\rightarrow 0$ . To solve $\lambda_{1,1}$ , we need the following lemma.

Lemma 3.1.

For sufficient small $\epsilon>0$ , consider

(3.12)

\begin{cases}\dot{\lambda}+\epsilon\frac{\lambda}{T-t}=(T-t)^{3}h,\\ \lambda(T)=0.\end{cases}

If $\|h\|_{0,\alpha}<\infty$ , (3.13) has a solution $\lambda$ , which can be write as

(3.13)

\lambda=-(T-t)^{\epsilon}\int_{t}^{T}(T-s)^{3-\epsilon}h(s)ds,

and there exists constant $C_{2}$ satisfies

(3.14)

\|\lambda\|_{1,\alpha}\leq C_{2}\|h\|_{0,\alpha}.

Proof: In fact, if $\epsilon$ is small enough, $\int_{t}^{T}(T-s)^{3-\epsilon}h(s)ds$ is integrable, then

(T-t)^{\epsilon}\int_{t}^{T}(T-s)^{3-\epsilon}h(s)ds

is well defined. One can check directly that (3.13) satisfies (3.12).
Then we check (3.14). By equation (3.12),

\dot{\lambda}=-\epsilon\frac{\lambda}{T-t}+(T-t)^{3}h,

where $(T-t)^{3}h$ is in $C^{\alpha}$ . By (3.13)

-\epsilon\frac{\lambda}{T-t}=\epsilon(T-t)^{\epsilon-1}\int_{t}^{T}(T-s)^{3-\epsilon}h(s)ds,

it’s easy to check it is in $C^{1}$ and

|\frac{d(T-t)^{\epsilon-1}\int_{t}^{T}(T-s)^{3-\epsilon}h(s)ds}{dt}|\leq\|h\|_{\infty},

then Lemma 3.1 follows immediately.

After finished the proof of Lemma 3.1. We may rewrite (3.11) as a linear equation about $\overline{\lambda_{1,1}}$

(3.15)

\begin{split}&\dot{\overline{\lambda}}_{1,1}\int_{R^{5}}W_{6}^{2}+\frac{2M\int_{R^{5}}U^{p}dy_{1}}{\kappa_{1}(T-t)}\overline{\lambda_{1,1}}+\delta_{R,T,\delta_{0}}(\dot{\lambda}_{1,0}+\dot{\lambda}_{1,1}+(T-t)\lambda_{1}^{\frac{1}{2}})\\ &-\int_{R^{5}}U(y_{1})^{p}dy_{1}M(T-t)\frac{\lambda_{1,1}^{2}}{2\lambda_{1,0}^{\frac{1}{2}}(\lambda_{1}^{\frac{1}{2}}+\lambda_{1,0}^{\frac{1}{2}})^{2}}=0.\end{split}

If M sufficient small, then for given $\lambda_{1,1}\in C^{1,\alpha}$ with $|\lambda_{1,1}|\leq\delta\lambda_{1,0},|\dot{\lambda}_{1,1}|\leq\delta|\dot{\lambda}_{1,0}|$ , $\delta$ small enough. By Lemma 3.1, we can solve $\overline{\lambda_{1,1}}$ with the following estimate

\|\overline{\lambda_{1,1}}\|_{1,\alpha}\leq C_{1}(\delta_{R,T,\delta_{0}}+\|\lambda_{1,1}\|_{1,\alpha}^{2})\lambda_{1,0}.

Due to our choice of R sufficiently large, T sufficiently small, $\delta_{0}$ sufficiently small and $\delta$ small, by the fixed point theorem we can solve $\lambda_{1}$ .

Similarly, we can solve $\lambda_{2}$ in a same way. As for $\xi_{i}=(\xi_{i,1},\cdots,\xi_{i,5})$ , and $\xi_{i,j}$ , defined by following equation

\int_{B_{2R}}H_{i}(y_{i},t)W_{j}(y_{i})dy_{i}=0.

That is, for $j=1,\cdots,5$ ,

\int_{B_{2R}}\lambda_{i}\dot{\xi}_{i}(\partial_{j}U(y_{i}))^{2}dy_{i}+\int_{B_{2R}}\lambda_{i}^{\frac{3}{2}}pU(y_{i})^{p-1}(-Z_{1}-Z_{2}+\psi(\xi_{i}+\lambda y_{i}))dy_{i}=0.

Because $Z_{1}$ , $Z_{2}$ are given, then for fixed $\psi$ , after solve $\lambda_{i}(\psi)$ we can obtain $\xi_{i,j}$ from the above equations.

4. The inner problem

In this section, we use Lemma 2.1 to obtain an estimate of $\psi_{1}$ and $\psi_{2}$ . Recall the inner problem

\begin{cases}\lambda_{1}^{2}\partial_{t}\phi_{1}=\Delta_{y_{1}}\phi_{1}+pU^{p-1}(y_{1})+H_{1}(\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2})\ \ \ (y_{1},t)\in\ B_{2R}(0)\times(0,T),\\ \phi_{1}(y,0)=\phi_{1,0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_{1}\in B_{2R}.\end{cases}

\begin{cases}\lambda_{2}^{2}\partial_{t}\phi_{2}=\Delta_{y_{2}}\phi_{2}+pU^{p-1}(y_{2})+H_{2}(\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2})\ \ \ (y_{2},t)\in\ B_{2R}(0)\times(0,T),\\ \phi_{2}(y,0)=\phi_{2,0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y_{2}\in B_{2R}.\end{cases}

Where

H_{i}(\psi,\lambda_{i},\xi_{i}):=\lambda_{i}^{\frac{3}{2}}pU(y_{i})^{p-1}(-Z_{1}(\xi_{i}+\lambda_{i}y_{i})-Z_{2}(\xi_{i}+\lambda_{i}y_{i})+\psi(\xi_{i}+\lambda_{i}y_{i}))+E_{i}(y_{i},t),

For given $\psi$ satisfies (2.5), then

H_{1}(\psi)\leq\lambda_{1}^{\frac{7}{4}}\frac{1}{1+|y_{1}|^{3}}.

H_{2}(\psi)\leq\lambda_{2}^{\frac{3}{2}}\frac{1}{1+|y_{2}|^{3}}.

Notice that for any $a\in(0,1)$ , the decay rate of $H_{i}(y_{i},t)$ is 2+a. Also due to the choice of $\lambda_{i}$ and $\xi_{i}$ , $H_{i}$ satisfies the orthogonality condition. Then by Lemma 2.1 we obtain

Proposition 4.1.

For any $a\in(0,1)$

\|\phi_{1}(y_{1},t)\|_{*a,\frac{7}{4}}\leq C_{3}\|H_{1}(y_{1},t)\|_{2+a,\frac{7}{4}}.

\|\phi_{2}(y_{2},t)\|_{*a,\frac{3}{2}}\leq C_{3}\|H_{2}(y_{1},t)\|_{2+a,\frac{3}{2}}.

The norm here means in the corresponding variable, that is

(4.1)

|\phi_{1}(y_{1},t)|\leq\lambda_{1}^{\frac{7}{4}}\frac{R^{6-a}}{1+|y_{1}|^{6}},\ \ \ \ |\phi_{2}(y_{2},t)|\leq\lambda_{2}^{\frac{3}{2}}\frac{R^{6-a}}{1+|y_{2}|^{6}}.

5. The outer problem

To solve outer problem, we also need to treat it as a linear problem

(5.1)

\begin{cases}\partial_{t}\psi_{*}&=\Delta_{x}\psi_{*}+g(\psi)\ \ \ \ (x,t)\in\ \mathbb{R}^{5}\times(0,T),\\ \psi_{*}(\cdot,0)&=\psi_{0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in\ \mathbb{R}^{5}\end{cases}

Our goal of this section is to prove $\psi$ has the following estimates

\begin{split}|\psi|\leq\begin{cases}\delta_{0}(T-t)(1+|z|^{4})\ &\ |z|\leq(T-t)^{-\frac{1}{4}},\\ \frac{\delta_{0}}{1+|x|^{2}}\ &\ |z|>(T-t)^{-\frac{1}{4}},\end{cases}\end{split}

where

\delta_{0}=\frac{\|Z_{2}\|_{\infty}}{10}.

Define space $B$ with the norm

\|\psi\|_{B}:=\sup_{x\in R^{5},t\in[0,T)}|\psi(x,t)|(T-t)^{-1}(1+|z|^{4}),\ \ \ \ z=\frac{x}{T-t},\ \ \ \ x\in\mathbb{R}^{5}.

From classic parabolic theorem, for given $\psi$ satisfies $\|\psi\|_{B}<\infty$ , equation (5.1) has a unique solution $\psi_{*}$ . Define

\psi_{*}=T[\psi].

5.1. Choosing of the parameters

We choose initial data of this form

(5.2)

\psi_{0}=(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{x}{e^{\frac{3}{8}\tau}}),

where $\boldsymbol{d}=(d_{\alpha_{0}},d_{\alpha_{1}}\cdots,d_{\alpha_{m}})$ is a parameter and $\boldsymbol{e}=(e_{\alpha_{0}},e_{\alpha_{1}}\cdots,e_{\alpha_{m}})$ is chosen such that $\boldsymbol{e}=(e_{\alpha_{0}},e_{\alpha_{1}}\cdots,e_{\alpha_{m}})$ are the eigenfunctions of $-A_{z}$ with eigenvalue less than or equal 2.
We use the self-similar variable

\varphi(z,\tau)=\psi_{*}(x,t),\ \ \ \ T-t=e^{-\tau}.\\

The function $\varphi(z,\tau)$ solves

(5.3)

\begin{split}\begin{cases}\varphi_{\tau}=\textit{A}\varphi+e^{-\tau}G(\phi,\psi,\lambda_{1},\lambda_{2},\xi_{1},\xi_{2})\ \ \ \ \ \ \ \ \ (z,\tau)\in\ \mathbb{R}^{5}\times(\tau_{0},\infty)\\ \varphi\mid_{\tau=\tau_{0}}=(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}}).\end{cases}\end{split}

We decompose the initial data in the eigenspace of $A$ , into two parts, $Y_{m}=span\{e_{\alpha_{0}},e_{\alpha_{1}}\cdots,e_{\alpha_{m}}\}$ and it’s orthogonal complement,

(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})=\sum d_{j}(e_{j}\eta(\frac{z}{e^{\frac{3}{8}\tau}}),e_{k})e_{k}+\{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})\}^{\perp},

especially

(\{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})\}^{\perp},e_{i})=0,\ \ \ \ i=1,\cdots,m,\ \ \ \tau=\tau_{0}.

We define $\Phi$ as

\varphi=\boldsymbol{b}(\tau)\cdot\boldsymbol{e}+\Phi.

where $\boldsymbol{b}(\tau)=(b_{\alpha_{1}}(\tau),\cdots,b_{\alpha_{m}}(\tau))$ . Choose $b_{\tau}$ as

(5.4)

b_{\alpha_{i}}(\tau)=-e^{-\lambda_{\alpha_{i}}\tau}\int_{\tau}^{\infty}e^{(\lambda_{\alpha_{i}}-1)\tau^{\prime}}(G,e_{\alpha_{i}})_{\rho}d\tau^{\prime}.

Then $\Phi$ satisfies

(5.5)

\Phi_{\tau}=\textit{A}_{z}\Phi+e^{-\tau}G-\sum_{i=1}^{m}\lambda_{\alpha_{i}}b_{\alpha_{i}}e_{\alpha_{i}}-\sum_{i=1}^{m}\frac{db_{\alpha_{i}}}{d\tau}e_{\alpha_{i}}:=\textit{A}_{z}\Phi+e^{-\tau}G^{\perp}.

By Lemma 5.1, it’s easy to see

(5.6)

|(G,e_{\alpha_{i}})|\lesssim e^{-\frac{7}{6}\tau}.

From the definition of $\boldsymbol{d}$ we have

(5.7)

(I+D)\boldsymbol{d}=\boldsymbol{b}(\tau_{0}),

where $D$ is a $(m+1)\times(m+1)$ matrix

(5.8)

D_{ij}=(e_{\alpha_{i}},(1-\eta(\frac{z}{e^{\frac{3}{8}\tau}})\mid_{\tau=\tau_{0}}e_{\alpha_{j}}).

For $\tau_{0}$ sufficient large, $|D_{ij}|<<1$ , then

(5.9)

|\boldsymbol{d}|\lesssim|\boldsymbol{b}(\tau)|\lesssim e^{-\frac{13}{6}\tau_{0}}.

Because $e_{\alpha}$ are given functions with eigenvalues less than or equal to 2, so

(5.10)

\begin{split}|b_{\alpha}e_{\alpha}|\lesssim e^{-\frac{13}{6}\tau}(1+|z|^{4}).\end{split}

5.2. The estimate of G

We give an estimate of G in this subsection. Let $R=e^{\frac{\tau_{0}}{2}}$ . Recall (2.11) $G$ could be written as

\begin{split}G=&\sum_{i=1,2}p\lambda_{i}^{-2}(1-\eta_{R,i})U(y_{i})^{p-1}(-Z_{1}\eta-Z_{2}+\psi)+\sum_{i=1,2}A[\phi_{i}]\\ &+\sum_{i=1,2}B[\phi_{i}]+\sum_{i=1,2}\lambda_{i}^{-\frac{7}{2}}E_{i}(1-\eta_{R,i})+N.\end{split}

Let $\textbf{1}_{z\in\Omega}(z)$ be a function on $\mathbb{R}^{5}$ defined by $\textbf{1}_{z\in\Omega}(z)=1$ if $z\in\Omega$ and $\textbf{1}_{z\in\Omega}(z)=0$ if $z\notin\Omega$ .

Lemma 5.1.

Under the above notation, we have

\begin{split}&|G|\\ &\lesssim\frac{\lambda_{1}^{-2}}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|x|}\leq 1}+\frac{\lambda_{2}^{-2}}{1+|y_{2}|^{2+a}}R^{-\frac{1}{2}}\textbf{1}_{{|x-q|}\leq 1}+e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}\\ &+e^{-\frac{7\tau}{6}}|z|^{4}\textbf{1}_{{|z|}\leq 2e^{\frac{\tau}{2}}}+e^{-\frac{7\tau}{3}}|z|^{10}\textbf{1}_{{|z|}\leq 2e^{\frac{\tau}{4}}}+e^{-\frac{7\tau}{3}}|z|^{6}\textbf{1}_{{|z|}\leq 2e^{\frac{3}{8}\tau}}+e^{-\frac{7}{3}\tau}\textbf{1}_{{|z|}\leq 1}\\ &+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{4}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|x|}\leq 1}+e^{-5\tau}|z|^{18}+e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}\\ &+e^{-\frac{7\tau}{6}}|z|^{4}\textbf{1}_{{|z|}\leq 2e^{\frac{\tau}{2}}}+e^{-\frac{7\tau}{3}}|z|^{10}\textbf{1}_{{|z|}\leq 2e^{\frac{\tau}{4}}}+e^{-\frac{7\tau}{3}}|z|^{6}\textbf{1}_{{|z|}\leq 2e^{\frac{3}{8}\tau}}\\ &+e^{-\frac{7}{3}\tau}\textbf{1}_{{|z|}\leq 1}+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{4}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}.\end{split}

Proof: We can estimate the terms of G one by one.
For $|z|\leq e^{\frac{\tau}{2}}$ , then

\begin{split}&\lambda_{1}^{-2}(1-\eta_{R,1})U(y_{1})^{p-1}(-Z_{1}\eta-Z_{2}+\psi)\\ &\lesssim\lambda_{1}^{-2}(1-\eta_{R,1})\frac{1}{y_{1}^{4}}(e^{-\tau}+e^{-\frac{\tau}{2}}|z|)\textbf{1}_{{|z|}\leq 1}+\lambda_{1}^{2}(1-\eta_{R,1})\frac{1}{y_{1}^{3}}e^{-\frac{7\tau}{2}}\textbf{1}_{1\leq|z|\leq e^{\frac{\tau}{2}}}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{y_{1}^{4}}(1-\eta_{R,1})e^{-\tau}\textbf{1}_{{|z|}\leq 1}+\lambda_{1}^{-2}\frac{1}{y_{1}^{3}}(1-\eta_{R,1})e^{-\frac{\tau}{2}}e^{-\frac{7\tau}{2}}\textbf{1}_{{|z|}\leq 1}+\lambda_{1}^{2}(1-\eta_{R,1})\frac{1}{y_{1}^{3}}e^{-\frac{7\tau}{2}}\textbf{1}_{1\leq|z|\leq e^{\frac{\tau}{2}}}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{{|z|}\leq 1}+\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{{|x|}\leq 1}\textbf{1}_{1\leq|z|\leq e^{\frac{\tau}{2}}}.\end{split}

If $|z|\geq e^{\frac{\tau}{2}}$ , then

\begin{split}\lambda_{1}^{-2}(1-\eta_{R,1})U(y_{1})^{p-1}(-Z_{1}\eta_{1}-Z_{2}+\psi)&\lesssim\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}.\end{split}

Because the control of $Z_{1}$ , $Z_{2}$ and $\psi$ , we deduce

\begin{split}&\lambda_{2}^{-2}(1-\eta_{R,2})U(y_{2})^{p-1}(-Z_{1}\eta_{1}-Z_{2}+\psi)\\ &\lesssim\lambda_{2}^{2}(1-\eta_{R,2})\frac{1}{y_{2}^{4}}(e^{-\tau}+e^{-\frac{\tau}{2}}|z|)\textbf{1}_{{|x-q|}\leq 1}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x-q|}\geq 1}\\ &\lesssim\lambda_{2}^{-2}(1-\eta_{R,2})\frac{1}{y_{2}^{4}}\textbf{1}_{{|x-q|}\leq 1}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x-q|}\geq 1}\\ &\lesssim\lambda_{2}^{-2}\frac{1}{1+|y_{2}|^{2+a}}R^{-1}\textbf{1}_{{|x-q|}\leq 1}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x-q|}\geq 1}.\\ \end{split}

In the self-similar variable

\lambda_{2}^{-2}(1-\eta_{R,2})\frac{1}{y_{2}^{4}}\textbf{1}_{{|x-q|}\leq 1}\lesssim e^{-4\tau}+e^{-5\tau}|z|^{18}.

From Proposition 4.1

|\phi_{1}(y_{1},t)|\leq C_{1}\lambda_{1}^{\frac{7}{4}}\frac{R^{6-a}}{1+|y_{1}|^{6}},\quad|\phi_{2}(y_{2},t)|\leq C_{2}\lambda_{2}^{\frac{3}{2}}\frac{R^{6-a}}{1+|y_{2}|^{6}},

therefore

\begin{split}A[\phi_{1}]+B[\phi_{1}]&=\lambda_{1}^{-\frac{7}{2}}[\Delta_{y_{1}}\eta_{R,1}\phi_{1}+2\nabla_{y_{1}}\eta_{R,1}\nabla_{y_{1}}\phi_{1}]\\ &\lesssim\lambda_{1}^{-2}e^{-\tau}R^{-2-a}\textbf{1}_{R\leq{|y_{1}|}\leq 2R}+\frac{e^{-8\tau}R^{6-a}}{\lambda_{1}^{2}(1+|y_{1}|^{6})}\textbf{1}_{{|y_{1}|}\leq 2R}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a-\frac{1}{2}}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{R\leq{|y_{1}|}\leq 2R}+\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{{|y_{1}|}\leq 2R}.\end{split}

Similarly

\begin{split}A[\phi_{2}]+B[\phi_{2}]&\lesssim\lambda_{2}^{-2}R^{-2-a}\textbf{1}_{R\leq{|y_{2}|}\leq 2R}+\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|y_{2}|}\leq 2R}\\ &\lesssim\lambda_{2}^{-2}\frac{1}{1+|y_{2}|^{2+a-\frac{1}{2}}}R^{-\frac{1}{2}}\textbf{1}_{R\leq{|y_{2}|}\leq 2R}+\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|y_{2}|}\leq 2R}\\ &\lesssim e^{-4\tau}+e^{-5\tau}|z|^{18}.\end{split}

Use the definition of $E_{1}$ , we have

\begin{split}\lambda_{1}^{-\frac{7}{2}}E_{1}(1-\eta_{R,1})&\lesssim\lambda_{1}^{-2}\lambda_{1}^{\frac{1}{2}}e^{\tau}\frac{1}{|y_{1}|^{3}}(1-\eta_{R,1})\textbf{1}_{|x|\leq 1}+\lambda_{1}^{-2}\lambda_{1}^{\frac{1}{2}}e^{\tau}\frac{1}{|y_{1}|^{3}}(1-\eta_{R,1})\textbf{1}_{|x|\geq 1}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{|x|\leq 1}+\frac{e^{-5\tau}}{1+|x|^{3}}\textbf{1}_{|x|\geq 1}.\end{split}

Similarly

\begin{split}\lambda_{2}^{-\frac{7}{2}}E_{2}(1-\eta_{R,2})&\leq\frac{\lambda_{2}^{-2}\lambda_{2}^{\frac{1}{2}}e^{\tau}}{|y_{2}|^{3}}(1-\eta_{R,2})\textbf{1}_{|x-q|\leq 1}+\lambda_{2}^{-2}\lambda_{2}^{\frac{1}{2}}e^{\tau}\frac{1}{|y_{2}|^{3}}(1-\eta_{R,2})\textbf{1}_{|x-q|\geq 1}\\ &\leq\frac{\lambda_{2}^{-2}}{1+|y_{2}|^{2+a}}R^{-1}\textbf{1}_{|x|\geq 1}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{|x-q|\geq 1}\\ &\leq e^{-4\tau}+e^{-5\tau}|z|^{18}+\frac{e^{-2\tau}}{1+|x|^{3}}.\end{split}

From the definition of $Z_{1}$ and $\eta_{1}$ , it’s easy to estimate the last part of $N$

\begin{split}|\frac{\partial\eta_{1}}{\partial t}Z_{1}+\nabla_{x}Z_{1}\nabla\eta_{1}-\Delta\eta_{1}Z_{1}|\lesssim e^{\frac{3}{4}\tau}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}\lesssim e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}.\end{split}

Then we deal with another part of $N$ . If $x$ is away from q,

\begin{split}&|U_{\lambda_{1},\xi_{1}}+U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2}|^{p-1}(U_{\lambda_{1},\xi_{1}}+U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2})\\ &\quad-U_{\lambda_{1},\xi_{1}}^{p}-U_{\lambda_{2},\xi_{2}}^{p}-pU_{\lambda_{1},\xi_{1}}^{p-1}(\theta-Z_{1}\eta_{1}-Z_{2})-pU_{\lambda_{2},\xi_{2}}^{p-1}(\theta-Z_{1}\eta_{1}-Z_{2})\\ &\lesssim U_{\lambda_{1}\xi_{1}}^{p-2}(U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2})^{2}+U_{\lambda_{2}\xi_{2}}^{p}+(U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2})^{p}\\ &\quad+U_{\lambda_{1}\xi_{1}}^{p-1}U_{\lambda_{2}\xi_{2}}.\end{split}

Because $\|\phi_{1}\|_{*a,\frac{7}{4}}$ is bounded

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-2}(\lambda_{1}^{-\frac{3}{2}}\phi_{1}(y_{1},t)\eta_{R}(y_{1}))^{2}&\lesssim\lambda_{1}^{-\frac{1}{2}}\frac{1}{1+|y_{1}|}\frac{e^{-2\tau}R^{12-2a}}{1+|y_{1}|^{12}}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{13}}e^{-8\tau}R^{12-2a}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{{|x|}\leq 1}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}.\end{split}

And we have

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-2}Z_{1}^{2}\eta_{1}^{2}&\lesssim\lambda_{1}^{-2}\frac{\lambda_{1}^{\frac{3}{2}}e^{-2\tau}(1+|y_{1}|^{1+a})}{1+|y_{1}|^{2+a}}\textbf{1}_{{|z|}\leq 1}+e^{-2\tau}\eta_{1}\textbf{1}_{{|z|}\geq 1}\\ &\lesssim\lambda_{1}^{-2}\frac{e^{-\frac{5}{2}\tau}}{1+|y_{1}|^{2+a}}\textbf{1}_{{|z|}\leq 1}+e^{-2\tau}\eta_{1}\textbf{1}_{{|z|}\geq 1}\\ &\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{{|z|}\leq 1}+e^{-2\tau}\eta_{1}\textbf{1}_{{|z|}\geq 1}.\end{split}

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-2}\Psi^{2}\lesssim\lambda_{1}^{-2}\frac{\lambda_{1}^{\frac{3}{2}}e^{-2\tau}(1+|y_{1}|^{1+a})}{1+|y_{1}|^{2+a}}\textbf{1}_{|z|\leq e^{\frac{\tau}{4}}}+\frac{e^{-\frac{19}{8}\tau}}{1+|x|^{4}}\textbf{1}_{|z|\geq e^{\frac{\tau}{4}}}\\ \lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}\textbf{1}_{|z|\leq e^{\frac{\tau}{4}}}+\frac{e^{-\frac{19}{8}\tau}}{1+|x|^{4}}\textbf{1}_{|z|\geq e^{\frac{\tau}{4}}}.\end{split}

Similarly for $|z|\leq 1$ ,

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-2}Z_{2}^{2}\lesssim\lambda^{-\frac{1}{2}}\frac{1}{1+|y_{1}|}e^{-\tau}&\lesssim\lambda_{1}^{-2}\frac{e^{-\frac{3}{2}\tau}}{1+|y_{1}|^{2+a}}\lesssim\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-1}.\end{split}

For $|z|\geq 1$ , and since $Z_{2}$ is a fast decay function,

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-2}Z_{2}^{2}\lesssim\frac{e^{-\frac{3}{2}\tau}}{1+|x|^{4}}.\end{split}

And $U_{\lambda_{1}\xi_{1}}^{p-2}U_{\lambda_{2}\xi_{2}}^{2}$ can be control by $U_{\lambda_{1}\xi_{1}}^{p-1}U_{\lambda_{2}\xi_{2}}$ and $U_{\lambda_{2}\xi_{2}}^{p}$ .

Moreover, for $|x|\leq 1$

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-1}U_{\lambda_{2}\xi_{2}}\lesssim\lambda_{1}^{-2}\frac{e^{-3\tau}}{1+|y_{1}|^{4}}\textbf{1}_{|x|\leq 1}.\end{split}

For $|x|\geq 1$ , $|x-q|\geq 1$ , we have

\begin{split}U_{\lambda_{1}\xi_{1}}^{p-1}U_{\lambda_{2}\xi_{2}}\lesssim\frac{e^{-17\tau}}{1+|x|^{4}}.\end{split}

\begin{split}(\lambda_{1}^{-\frac{3}{2}}\phi_{1}(y_{1},t)\eta_{R}(y_{1}))^{p}&\lesssim\lambda_{1}^{-2}\frac{e^{-8\tau}e^{-\frac{7}{3}\tau}R^{12}}{1+|y_{1}|^{12}}\lesssim\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}.\end{split}

\begin{split}(\lambda_{2}^{-\frac{3}{2}}\phi_{2}(y_{2},t)\eta_{R}(y_{2}))^{p}\lesssim\lambda_{2}^{-2}\frac{e^{-4\tau}R^{12}}{1+|y_{2}|^{12}}.\end{split}

As before

\begin{split}|\Psi|^{p}+|Z_{1}\eta_{1}|^{p}&\lesssim e^{-\frac{7\tau}{3}}(1+|z|^{\frac{28}{3}})\textbf{1}_{|z|\leq e^{\frac{\tau}{4}}}+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{\frac{14}{3}}}\textbf{1}_{|z|\geq e^{\frac{\tau}{4}}}+e^{-\frac{7\tau}{3}}(1+|z|^{\frac{14}{3}})\eta_{1}\\ &\lesssim e^{-\frac{7\tau}{3}}(1+|z|^{10})\textbf{1}_{|z|\leq e^{\frac{\tau}{4}}}+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{|z|\geq e^{\frac{\tau}{4}}}+e^{-\frac{7\tau}{3}}(1+|z|^{6})\eta_{1}.\end{split}

Moreover

\begin{split}|Z_{2}|^{p}&\leq e^{-\frac{7\tau}{6}}|z|^{\frac{7}{3}}\textbf{1}_{|z|\leq e^{\frac{\tau}{2}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{|z|\geq e^{\frac{\tau}{2}}}\leq e^{-\frac{7\tau}{6}}|z|^{4}\textbf{1}_{|z|\leq e^{\frac{\tau}{2}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{|z|\geq e^{\frac{\tau}{2}}}.\end{split}

If $|x-q|\leq 1$

\begin{split}&|U_{\lambda_{1},\xi_{1}}+U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2}|^{p-1}(U_{\lambda_{1},\xi_{1}}+U_{\lambda_{2},\xi_{2}}+\theta-Z_{1}\eta_{1}-Z_{2})\\ &\quad-U_{\lambda_{1},\xi_{1}}^{p}-U_{\lambda_{2},\xi_{2}}^{p}-pU_{\lambda_{1},\xi_{1}}^{p-1}(\theta-Z_{1}\eta_{1}-Z_{2})-pU_{\lambda_{2},\xi_{2}}^{p-1}(\theta-Z_{1}\eta_{1}-Z_{2})\\ &\leq U_{\lambda_{2},\xi_{2}}^{p-2}(U_{\lambda_{1},\xi_{1}}+\theta-Z_{1}\eta_{1}-Z_{2})^{2}+U_{\lambda_{1},\xi_{1}}^{p}+(U_{\lambda_{1},\xi_{1}}+\theta-Z_{1}\eta_{1}-Z_{2})^{p}\\ &\quad+U_{\lambda_{2},\xi_{2}}^{p-1}U_{\lambda_{1},\xi_{1}}.\end{split}

And

\begin{split}U_{\lambda_{2},\xi_{2}}^{p-2}(U_{\lambda_{1},\xi_{1}}+\theta-Z_{1}\eta_{1}-Z_{2})^{2}\lesssim\frac{1}{\lambda_{2}^{\frac{1}{2}}(1+|y_{2}|)}\lesssim\frac{e^{-\frac{\tau}{2}}}{\lambda_{2}^{2}(1+|y_{2}|^{2+a})}\lesssim\frac{1}{\lambda_{2}^{2}(1+|y_{2}|^{2+a})}R^{-\frac{1}{2}}.\end{split}

Also

\begin{split}U_{\lambda_{1},\xi_{1}}^{p}+U_{\lambda_{2},\xi_{2}}^{p-1}U_{\lambda_{1},\xi_{1}}\lesssim e^{-14\tau}+\frac{e^{-6\tau}}{\lambda_{2}^{2}(1+|y|_{2}^{4})}\lesssim e^{-14\tau}+\lambda_{2}^{-2}\frac{1}{1+|y_{2}|^{2+a}}R^{-\frac{1}{2}}.\end{split}

In conclusion, we complete the proof.

By using Lemma 5.1, we can easily deduce that

Proposition 5.2.

The function $G$ given by (2.11) satisfies

(5.11)

\|G\|_{\rho}\lesssim e^{-\frac{7}{6}\tau}.

Proof: Recall $y_{1}=\frac{x-\xi_{1}}{\lambda_{1}}$ , $z=\frac{x}{\sqrt{T-t}}$ , that is

z=\frac{(y_{1}\lambda_{1}+\xi_{1})}{\sqrt{T-t}},

then we have

\begin{split}&|\int_{R^{5}}(\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{|x|\leq 1})^{2}e^{-\frac{|z|^{2}}{4}}dz|\\ &=\int_{R^{5}}(\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}})^{2}e^{-\frac{|z|^{2}}{4}}(\frac{\lambda_{1}}{\sqrt{T-t}})^{5}dy_{1}\\ &\lesssim\lambda_{1}e^{-2\tau}e^{-\frac{5}{2}\tau}\\ &\lesssim e^{-\frac{7}{2}\tau}.\end{split}

And

|\int_{R^{5}}(e^{-5\tau}|z|^{18})^{2}e^{-\frac{|z|^{2}}{4}}dz|\lesssim e^{-10\tau}.

For $\tau\geq\tau_{0}$ sufficient large, we have

\begin{split}|\int_{R^{5}}(e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\lesssim 2e^{\frac{3}{8}\tau}})^{2}e^{-\frac{|z|^{2}}{4}}dz|&\lesssim e^{\frac{3}{2}\tau}e^{-\frac{e^{\frac{3}{4}\tau}}{4}}\leq e^{-5\tau}.\end{split}

|\int_{R^{5}}(e^{-\frac{7\tau}{6}}|z|^{4}\textbf{1}_{{|z|}\lesssim e^{\frac{\tau}{2}}})^{2}e^{-\frac{|z|^{2}}{4}}dz|\leq e^{-\frac{7\tau}{3}}.

|\int_{R^{5}}(e^{-\frac{7\tau}{3}}|z|^{10}\textbf{1}_{{|z|}\lesssim e^{\frac{\tau}{4}}})^{2}e^{-\frac{|z|^{2}}{4}}dz|\leq e^{-\frac{14\tau}{3}}.

|\int_{R^{5}}(e^{-\frac{7\tau}{3}}|z|^{6}\textbf{1}_{{|z|}\lesssim e^{\frac{3}{8}\tau}})^{2}e^{-\frac{|z|^{2}}{4}}dz|\leq e^{-\frac{14\tau}{3}}.

|\int_{R^{5}}(e^{-\frac{7}{3}\tau}\textbf{1}_{{|z|}\leq 1})^{2}e^{-\frac{|z|^{2}}{4}}dz|\lesssim e^{-\frac{14\tau}{3}}.

Since the function $s^{3}e^{-\frac{s^{2}}{8}}$ is bounded and for $\tau\geq\tau_{0}$ sufficient large, we have

\begin{split}|\int_{R^{5}}(\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}})^{2}e^{-\frac{|z|^{2}}{4}}dz|&\lesssim\int_{e^{\frac{\tau}{2}}}^{\infty}s^{4}e^{-\frac{s^{2}}{4}}ds\\ &\lesssim\int_{e^{\frac{\tau}{2}}}^{\infty}s^{3}e^{-\frac{s^{2}}{8}}se^{-\frac{s^{2}}{8}}ds\\ &\lesssim e^{-\frac{e^{\tau}}{8}}\\ &\lesssim e^{-5\tau}.\end{split}

Similarly

\begin{split}|\int_{R^{5}}(\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}})^{2}e^{-\frac{|z|^{2}}{4}}dz|\lesssim e^{-5\tau}.\end{split}

Therefore

\|G\|_{\rho}\lesssim e^{-\frac{7}{6}\tau}.

5.3. The solving of $\Phi_{1}$

We divide $\Phi$ into two parts, one part comes from the initial data, another comes from nonhomogeneous term, consider

(5.12)

\begin{split}\begin{cases}\partial_{\tau}\Phi_{1}=A_{z}\Phi_{1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (z,\tau)\in\mathbb{R}^{5}\times[\tau_{0},\infty),\\ \Phi_{1}\mid_{\tau=\tau_{0}}=\{{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})}\}^{\perp}.\end{cases}\end{split}

Lemma 5.3.

There exists constant $K_{1}$ , such that

(5.13)

|\Phi_{1}(z,\tau)|\leq K_{1}e^{-2\tau}(1+|z|^{4}),\ \ \ \ \tau>\tau_{0}+1.

Proof: First we estimate the initial data

\{{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})}\}^{\perp}={(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})}-\sum d_{\alpha_{j}}(e_{\alpha_{j}}\eta(\frac{z}{e^{\frac{3}{8}\tau}}),e_{\alpha_{k}})_{\rho}e_{\alpha_{k}},

As in the discuss of last section, we can deduce that

|\{{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{-\frac{3}{8}\tau}})}\}^{\perp}|\lesssim e^{-\frac{13}{6}\tau_{0}}+e^{-\frac{13}{6}\tau_{0}}|z|^{4},

and

\begin{split}\|\{{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})}\}^{\perp}\|_{\rho}\lesssim e^{-\frac{13}{6}\tau_{0}}.\\ \end{split}

We easily deduced that

\begin{split}\frac{1}{2}\partial_{\tau}\|\Phi_{1}\|_{\rho}^{2}=(A_{z}\Phi_{1},\Phi_{1})\leq-\frac{5}{2}\|\Phi_{1}\|_{\rho}^{2}.\end{split}

witch is

\partial_{\tau}(\|\Phi_{1}\|_{\rho}^{2}e^{5\tau})\leq 0.

Integrating it from $\tau_{0}$ to $\tau$

\begin{split}\|\Phi_{1}\|_{\rho}\leq\|{(\boldsymbol{d}\cdot\boldsymbol{e})\eta(\frac{z}{e^{\frac{3}{8}\tau}})}^{\perp}\|_{\rho}e^{-\frac{5}{2}(\tau-\tau_{0})}\lesssim e^{-2\tau}.\end{split}

Let $r_{0}$ be the constant such that in $|z|\geq r_{0}$ , $e_{2}>0$ . From the classical $L^{2}$ estimate of parabolic equation and the definition of $\rho$ we have, for $\tau>\tau_{0}+1,|z|<r_{0}$

|\Phi_{1}|\lesssim\|\Phi_{1}\|_{2,B_{2r_{0}}}\lesssim e^{-2\tau}.

Now we construct a comparison function $K_{1}e^{-2\tau}e_{2}(z)$ . In fact, $K$ sufficient large, we prove it is a super solution. When $|z|=r_{0}$

|\Phi_{1}|<K_{1}e^{-2\tau}e_{2}(z).\ \

For $\tau_{0}\leq\tau\leq\tau_{0}+1$ , by Lemma 2.3,

|\Phi_{1}|\lesssim e^{-\frac{13}{6}\tau_{0}}+e^{-\frac{13}{6}\tau_{0}}e^{-2(\tau-\tau_{0})}|z|^{4}.

Therefore when $\tau=\tau_{0}+1$

|\Phi_{1}|\lesssim e^{-\frac{13}{6}\tau_{0}}+e^{-\frac{13}{6}\tau_{0}}|z|^{4}\leq K_{1}e^{-2\tau}e_{2}(z).

We can easily verify

\partial_{\tau}(Ke^{-2\tau}e_{2}(z))-A_{z}(Ke^{-2\tau}e_{2}(z))=0=\partial_{\tau}\Phi_{1}-A_{z}\Phi_{1}.

By comparison principle,

|\Phi_{1}|<K_{1}e^{-2\tau}e_{2}(z).

5.4. The solving of $\Phi_{2}$

Recall the equation of $\Phi_{2}$

(5.14)

\begin{cases}\partial_{t}\Phi_{2}-\Delta\Phi_{2}=G^{\bot}\ \ \ \ \ \ \ \ \ \ \ (z,\tau)\in\mathbb{R}^{5}\times[\tau_{0},\infty),\\ \Phi_{2}\mid_{\tau=\tau_{0}}=0.\end{cases}

Our goal is to prove $\Phi_{2}$ vanishes around 0 and $\|\Phi_{2}\|_{\infty}$ small. So we first prove the following lemma.

Lemma 5.4.

Let $A_{z}=\Delta_{z}-\frac{z}{2}\cdot\nabla$ be the operator as before, then for any $\tau\geq\tau_{1}\geq\tau$ we have

(5.15)

|\int_{\tau_{1}}^{\tau}e^{A_{z}(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}|\lesssim\frac{e^{-\tau_{1}}}{R^{\frac{1}{2}}}+e^{-6\tau}|z|^{18}+e^{-\frac{7}{4}\tau}|z|^{4}+e^{-\frac{10}{3}\tau}|z|^{10}.

Proof: Notice

\begin{split}G^{\bot}=G-(G,e_{\alpha_{i}})e_{\alpha_{i}},\end{split}

and

\begin{split}|b_{\alpha}e_{\alpha}|\lesssim e^{-\frac{13}{6}\tau}(1+|z|^{4}).\end{split}

Then

\begin{split}G\lesssim&\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|x|}\leq 1}+e^{-5\tau}|z|^{18}+e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}\\ &+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{4}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}+e^{-\frac{7\tau}{3}}(1+|z|^{10}).\\ \end{split}

We estimate the following terms

\begin{split}\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{4}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}\leq e^{-\tau}|z|^{4}.\end{split}

Let $\Psi_{1}$ be the solution of following equation

\begin{cases}\partial_{t}\Psi_{1}-\Delta\Psi_{1}=\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|x|}\leq 1}\ \ \ \ \ \ \ \ \ \ \ (x,t)\in\mathbb{R}^{5}\times[0,T),\\ \Psi_{1}\mid_{t=0}=0.\end{cases}

By Lemma 2.4, we obtain

|\Psi_{1}|\lesssim\frac{e^{-\tau_{1}}}{R^{\frac{1}{2}}}.

$\Psi_{2}$ is the solution of following equation

\begin{cases}\partial_{t}\Psi_{2}-\Delta\Psi_{1}=&e^{-5\tau}|z|^{18}+e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{4}}}\\ &+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}}+e^{-\frac{7\tau}{3}}(1+|z|^{10})\ \ \ \ \ \ \ \ \ \ \ (x,t)\in\mathbb{R}^{5}\times[0,T),\\ \Psi_{2}\mid_{t=0}=0.\end{cases}

By Duhamel’s principle

\begin{split}|\Psi_{2}|\leq&\int_{\tau_{1}}^{\tau}e^{-\frac{10}{3}\tau^{\prime}}(1+e^{-5(\tau-\tau^{\prime})}|z|^{10})+e^{-6\tau^{\prime}}(1+e^{-9(\tau-\tau^{\prime})}|z|^{18})\\ &+e^{-\frac{7}{4}\tau^{\prime}}(1+e^{-2(\tau-\tau^{\prime}}|z|^{4})e^{-\frac{13}{6}\tau^{\prime}}(1+e^{-2(\tau-\tau^{\prime})}|z|^{4})d\tau^{\prime}\\ \lesssim&e^{-\frac{7}{4}\tau_{1}}+e^{-6\tau}|z|^{18}+e^{-\frac{7}{4}\tau}|z|^{4}+e^{-\frac{10}{3}\tau}|z|^{10}.\end{split}

Together with $\Psi_{1}$ , we obtain the conclusion.

Next we improve the estimate of Lemma 5.4 when $\tau$ is large

Lemma 5.5.

Under the assumptions of Lemma 5.4, we have

(5.16)

\begin{split}|\int_{\tau_{0}}^{\tau}e^{A_{z}(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|&\lesssim\frac{e^{-\tau}}{R^{\frac{1}{2}}}+e^{-\frac{13}{6}\tau}e^{\frac{5}{4}\tau_{0}}+e^{-\frac{7}{4}\tau}|z|^{4}+\frac{e^{-\tau}}{R^{\frac{1}{2}}}|z|^{2}\\ &\quad+e^{-6\tau}|z|^{18}+e^{-\frac{10}{3}\tau}|z|^{10}.\end{split}

Proof: We separate space-time into four cases:

(5.17)

\begin{split}&1.\ z\in\mathbb{R}^{5},\ \tau\in(\tau_{0},\tau_{0}+1).\\ &2.\ |z|\leq 4,\ \tau\in(\tau_{0}+1,\infty).\\ &3.\ 4\leq|z|\leq e^{\frac{\tau-\tau_{0}}{2}},\ \tau\in(\tau_{0}+1,\infty).\\ &4.\ |z|>e^{\frac{\tau-\tau_{0}}{2}},\ \tau\in(\tau_{0}+1,\infty).\end{split}

•

$z\in\mathbb{R}^{5},\tau\in(\tau_{0},\tau_{0}+1)$ , we conclude by Lemma 5.4.

•

$|z|\leq 4$ . Firstly we times $\Phi_{2}$ and integral with weight $\rho$ in (5.14)

\frac{1}{2}\frac{d\|\Phi_{2}\|_{\rho}^{2}}{d\tau}=(A_{z}\Phi_{2},\Phi_{2})_{\rho}+e^{-\tau}(G^{\perp},\Phi_{2})_{\rho}.

Since $(A_{z}\Phi_{2},\Phi_{2})_{\rho}\leq-\frac{5}{2}\|\Phi_{2}\|^{2}$ , we deduce from Lemma 5.1 that

(5.18)

\|\Phi_{2}\|_{\rho}\lesssim e^{-\frac{13}{6}\tau}.

Next, from Lemma 2.2 and the definition of $G^{\perp}$

\begin{split}|\int_{\tau_{0}}^{\tau-1}e^{A_{z}(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|&=|\int_{\tau_{0}}^{\tau-1}e^{A_{z}\frac{1}{2}}e^{A_{z}(\tau-\tau^{\prime}-\frac{1}{2})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|\\ &\lesssim\int_{\tau_{0}}^{\tau-1}\frac{\frac{e^{-\frac{1}{2}|z|^{2}}}{e^{4(2-e^{-\tau_{0}}+e^{-\tau_{0}-\frac{1}{2}})}}}{(e^{-\tau_{0}}-e^{-\tau_{0}-1})^{\frac{5}{4}}}e^{-\frac{5}{2}(\tau-\tau^{\prime})}\|e^{-\tau^{\prime}}G^{\bot}\|_{\rho}d\tau^{\prime}\\ &\lesssim\int_{\tau_{0}}^{\tau-1}e^{\frac{5}{4}\tau_{0}}e^{-\frac{5}{2}(\tau-\tau^{\prime})}e^{-\frac{13}{6}\tau^{\prime}}d\tau^{\prime}\\ &\lesssim e^{-\frac{13}{6}\tau}e^{\frac{5}{4}\tau_{0}}.\\ \end{split}

The integral from $\tau-1$ to $\tau$ can be estimated by Lemma 5.4, then

\begin{split}|\int_{\tau-1}^{\tau}e^{Az(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|\lesssim\frac{e^{-\tau}}{R^{\frac{1}{2}}}.\end{split}

•

$4\leq|z|\leq e^{\frac{\tau-\tau_{0}}{2}},\tau>\tau_{0}+1$ . Let $\tau_{2}$ define by $|z|=e^{\frac{\tau-\tau_{2}}{2}}$ , from the definition, $\tau>\tau_{2}$ , a similarly calculation shows

\begin{split}|\int_{\tau_{0}}^{\tau_{2}}e^{Az(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|&\leq|\int_{\tau_{0}}^{\tau_{2}}e^{\frac{5}{4}\tau_{0}}e^{-\frac{5}{2}(\tau-\tau^{\prime})}e^{-\frac{13}{6}\tau^{\prime}}G^{\bot}\tau^{\prime}|\\ &\lesssim e^{\frac{1}{3}\tau_{2}}e^{-\frac{5}{2}\tau}e^{\frac{5}{4}\tau_{0}}\\ &\lesssim e^{-\frac{13}{6}\tau}e^{\frac{5}{4}\tau_{0}},\\ |\int_{\tau_{2}}^{\tau-1}e^{Az(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|&\leq\frac{e^{-\tau_{2}}}{R^{\frac{1}{2}}}+e^{-6\tau}|z|^{18}+e^{-\frac{7}{4}\tau}|z|^{4}+e^{-\frac{10}{3}\tau}|z|^{10}\\ &\lesssim\frac{e^{-\tau}}{R^{\frac{1}{2}}}|z|^{2}+e^{-6\tau}|z|^{18}+e^{-\frac{7}{4}\tau}|z|^{4}+e^{-\frac{10}{3}\tau}|z|^{10}.\\ \end{split}

•

$|z|>e^{\frac{\tau-\tau_{0}}{2}}$ . Similar as before, by Lemma 5.4 and $|z|>e^{\frac{\tau-\tau_{0}}{2}}$ .

\begin{split}&|\int_{\tau_{0}}^{\tau}e^{Az(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|\\ &\lesssim\frac{e^{-\tau_{0}}}{R^{\frac{1}{2}}}+e^{-6\tau}|z|^{18}+e^{-\frac{7}{4}\tau}|z|^{4}+e^{-\frac{10}{3}\tau}|z|^{10}\\ &\lesssim\frac{e^{-\tau}}{R^{\frac{1}{2}}}|z|^{2}+e^{-6\tau}|z|^{18}+e^{-\frac{7}{4}\tau}|z|^{4}+e^{-\frac{10}{3}\tau}|z|^{10}.\end{split}

Combining all these estimates we complete the proof of Lemma 5.5.

Recall our goal is to show $\Phi_{*}(x,t)$ lays in space $X$ and it’s norm is smaller than $\delta_{0}$ . In the estimate of Lemma 5.5, when $\tau>\frac{15}{14}\tau_{0}$ the term

e^{-\frac{13}{6}\tau}e^{\frac{5}{4}\tau_{0}}\in X,

and

e^{-\frac{13}{6}\tau}e^{\frac{5}{4}\tau_{0}}<\delta_{0}e^{-\tau}.

But when $\tau$ is near $\tau_{0}$ we can not obtain the same estimate. Thus we need to employ Lemma 5.4. In Lemma 5.4, let $\tau_{1}=\tau_{0}$ , because $R=e^{\frac{\tau_{0}}{2}}$ , the first term in Lemma 5.4 is

\frac{e^{-\tau_{0}}}{R^{\frac{1}{2}}}={e^{-\tau_{0}}}e^{-\frac{\tau_{0}}{4}}={e^{-\frac{5}{4}\tau_{0}}}.

This is to say when $\tau<\frac{5}{4}\tau_{0}$ we have

(5.19)

|\Phi_{2}|<\delta_{0}e^{-\tau}.

So, in $|z|\leq e^{\frac{\tau}{4}}$ , for any $\tau\geq\tau_{0}$

(5.20)

|\Phi_{2}|<\delta_{0}e^{-\tau}.

5.5. A point wise estimate for $\psi$ in $|z|>e^{\frac{\tau}{4}}$

The estimate of last section is locally. In this section we make use of comparison principle to obtain an estimate in larger region.

Lemma 5.6.

There exists constant $K_{2}$ such that in the region $|z|>e^{\frac{\tau}{4}}$

(5.21)

|\psi|\leq K_{2}T^{\frac{1}{4}}.

Proof: We separate $\psi$ into 2 parts. Let $\Psi_{3}$ be the solution satisfies

\begin{cases}\partial_{t}\Psi_{3}-\Delta\Psi_{3}=\lambda_{2}^{-2}\frac{1}{1+|y_{2}|^{2+a}}R^{-\frac{1}{2}}\ \ \ \ \ \ \ \ \ \ \ (x,t)\in\mathbb{R}^{5}\times[0,T).\\ \Psi_{3}\mid_{t=0}=0,\end{cases}

Denote

\begin{split}G^{\prime}=&\lambda_{1}^{-2}\frac{1}{1+|y_{1}|^{2+a}}e^{-\tau}R^{-\frac{1}{2}}\textbf{1}_{{|x|}\leq 1}+e^{-\frac{3}{4}\tau}|z|^{4}\textbf{1}_{e^{\frac{3}{8}\tau}\leq{|z|}\leq 2e^{\frac{3}{8}\tau}}+e^{-\frac{7\tau}{6}}|z|^{4}\textbf{1}_{{|z|}\leq 2e^{\frac{\tau}{2}}}\\ &+e^{-\frac{7\tau}{3}}|z|^{10}\textbf{1}_{{|z|}\leq 2e^{\frac{\tau}{4}}}+e^{-\frac{7\tau}{3}}|z|^{6}\textbf{1}_{{|z|}\leq 2e^{\frac{3}{8}\tau}}+e^{-\frac{7}{3}\tau}\textbf{1}_{{|z|}\leq 1}\\ &+\frac{\delta_{0}^{\frac{7}{3}}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{4}}}+\frac{\|Z_{2}\|_{\infty}^{p}}{1+|x|^{4}}\textbf{1}_{{|z|}\geq e^{\frac{\tau}{2}}}+\frac{e^{-2\tau}}{1+|x|^{3}}\textbf{1}_{{|x|}\geq 1}.\end{split}

Let $\Psi_{4}$ satisfies

\begin{split}\begin{cases}\partial_{t}\Psi_{4}-\Delta\Psi_{4}=G^{\prime}\ \ \ \ \ \ \ \ \ \ \ (x,t)\in\mathbb{R}^{5}\times[0,T),\\ \Psi_{4}\mid_{t=0}={(d\cdot e)\eta(\frac{z}{e^{\frac{3}{8}\tau}})}.\end{cases}\end{split}

Use Lemma 2.4 we obtain

(5.22)

|\Psi_{3}|\leq T^{\frac{1}{4}}.

Then we check that $\overline{\Psi}=K_{2}(2e^{-\frac{1}{4}\tau_{0}}-e^{-\frac{1}{4}\tau})$ , $K_{2}$ sufficient large, is a super solution of $\Psi_{4}$ in $|z|>e^{\frac{\tau}{4}}$ .
First, from Lemma 5.3, (5.20) and

|b_{\alpha}e_{\alpha}|\lesssim e^{-\frac{13}{6}\tau}(1+|z|^{4}),

we obtain when $|z|=e^{\frac{\tau}{4}}$

|\psi|\lesssim e^{-\tau}.

Also $\psi=\Psi_{3}+\Psi_{4}$ , because $|\Psi_{3}|\leq T^{\frac{1}{4}}$ , for $|z|=e^{\frac{\tau}{4}}$

|\Psi_{4}|\leq e^{-\frac{1}{4}\tau_{0}}\leq\overline{\Psi},

and for $|z|>e^{\frac{\tau}{4}}$ , we verify that

\overline{\Psi}_{\tau}-A_{z}\overline{\Psi}=\frac{K_{2}}{4}e^{-\frac{1}{4}\tau}>e^{-\tau}G^{\prime}>\partial_{\tau}\Psi_{4}-A_{z}\Psi_{4}.

For $\tau=\tau_{0}$

|\Psi_{4}|\leq e^{-\frac{13}{6}\tau_{0}}e^{\frac{3}{2}\tau_{0}}\leq e^{-\frac{2}{3}\tau_{0}}\leq|\overline{\Psi}|.

Use comparison principle $|\psi|\leq\Psi_{3}+\Psi_{4}$ , we finished the proof.

Now Lemma 5.6 provides an $L^{\infty}$ estimate, then we give the decay estimate.

Lemma 5.7.

For $|x|\geq|q|+1$ , there exists constant $K_{3}>0$ such that

(5.23)

|\psi|\leq K_{3}(\frac{T^{3}-(T-t)^{3}}{|x|^{3}}+\frac{\|Z_{2}\|_{\infty}^{p}}{|x|^{2}}).

Proof: We are going to check that

\widetilde{\psi}=K_{3}(\frac{T^{6}-(T-t)^{6}}{|x|^{3}}+\frac{\|Z_{2}\|_{\infty}^{p}}{|x|^{2}}),

for $K_{3}$ sufficient large, is a super solution of $\psi$ in $|x|\geq|q|+1$ . First we verify that if $K_{3}$ large

\widetilde{\psi}_{t}-\Delta\widetilde{\psi}=K_{3}(\frac{3(T-t)^{2}}{|x|^{3}}+\frac{2\|Z_{2}\|_{\infty}^{p}}{|x|^{4}})\geq\psi_{t}-\Delta\psi.

From Lemma 5.6, for $|x|=q+1$

|\psi|\leq K_{2}T^{\frac{1}{4}}\leq\widetilde{\psi}.

When $t=0$ , because the intersection of $|x|\geq|q|+1$ and

\psi\mid_{t=0}={(d\cdot e)\eta(\frac{z}{e^{\frac{3}{8}\tau}})}

is empty. So in $|x|\geq|q|+1$

\psi\mid_{t=0}=0\leq\frac{\|Z_{2}\|_{\infty}^{p}}{|x|^{2}}=\widetilde{\psi}\mid_{t=0}.

Use comparison principle, we finished the proof.

6. The proof of theorem 1.1

After the preparation of the last few sections, we start to prove Theorem 1.1.
Because for $1\leq i\leq m$

\begin{split}|b_{\alpha_{i}}e_{\alpha_{i}}|\lesssim e^{-\frac{13}{6}\tau}(1+|z|^{4}),\end{split}

from Lemma 5.3

|\Phi_{1}(z,\tau)\leq Ce^{-2\tau}(1+|z|^{4}),\quad\tau\geq\tau_{0}.

combine with Lemma 5.5 and (5.20), we have

(6.1)

\begin{split}|\Phi_{2}|=&|\int_{\tau_{0}}^{\tau}e^{A_{z}(\tau-\tau^{\prime})}e^{-\tau^{\prime}}G^{\bot}d\tau^{\prime}|\leq\delta_{0}e^{-\tau}+e^{-\frac{7}{4}\tau}|z|^{4}+\frac{e^{-\tau}}{R^{\frac{1}{2}}}|z|^{2}\\ &+e^{-6\tau}|z|^{18}+e^{-\frac{10}{3}\tau}|z|^{10}.\end{split}

When $|z|\leq e^{-\frac{\tau}{4}}$

(6.2)

\begin{split}|\Phi_{2}|&\lesssim\delta_{0}e^{-\tau}+e^{-\frac{7}{4}\tau}|z|^{4}+\frac{e^{-\tau}}{R^{\frac{1}{2}}}|z|^{2}+e^{-6\tau}|z|^{18}+e^{-\frac{10}{3}\tau}|z|^{10}\\ &\lesssim\delta_{0}e^{-\tau}+e^{-\frac{7}{4}\tau}|z|^{4}+\frac{e^{-\tau}}{R^{\frac{1}{2}}}|z|^{2}+e^{-\frac{5}{2}\tau}|z|^{4}+e^{-\frac{11}{6}\tau}|z|^{4}.\\ \end{split}

If $\tau_{0}$ sufficient large, we have

(6.3)

|\Phi_{2}|\leq\delta_{0}e^{-\tau}(1+|z|^{4}),

and from Lemma 5.6 and Lemma 5.7 we have when $|z|\geq e^{-\frac{\tau}{4}}$

(6.4)

|\psi|\leq\delta_{0}\frac{1}{1+|x|^{2}}.

We know operator $T(\Phi)=\Phi_{*}$ maps X in to X, and from local Holder estimate and Ascoli-Arezela theorem we know $T$ is compact. By Schauder fixed point theorem, we know that $T$ has a fixed point $\Phi$ in X. Which finished the proof of Theorem 1.1

References

[1] C. Collot,Nonradial type II blow-up for the energy-supercritical semilinear heat equation. Anal. PDE 10 (2017), no. 1, 127–252.
[2] COLLOT C, MERLE F, RAPHAëL P.Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions. Comm. Math. Phys. 352 (2017), no. 1, 215–285.
[3] CORTáZAR C, DEL PINO M, MUSSO M.Green’s function and infinite-time bubbling in the critical nonlinear heat equation. J. Eur. Math. Soc. (JEMS) 22 (2020), no. 1, 283–344.
[4] DáVILA J, DEL PINO M, WEI J.Singularity formation for the two-dimensional harmonic map flow into $\mathbb{S}^{2}$ .Invent. Math. 219 (2020), no. 2, 345–466.
[5] DEL PINO M, MUSSO M, WEI J.Geometry driven type ii higher dimensional blow-up for the critical heat equationarXiv:1710.11461, 2017.
[6] DEL PINO M, MUSSO M, WEI J C.Type II blow-up in the 5-dimensional energy critical heat equation. Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 1027–1042.
[7] DEL PINO M, MUSSO M, WEI J.Infinite-time blow-up for the 3-dimensional energy-critical heat equation. Anal. PDE 13 (2020), no. 1, 215–274.
[8] DEL PINO M, MUSSO M, WEI J, et al.Type ii finite time blow-up for the three dimensional energy critical heat equationarXiv:2002.05765, 2020.
[9] DEL PINO M, MUSSO M, WEI J, et al.Type II finite time blow-up for the energy critical heat equation in $\mathbb{R}^{4}$ .Discrete & Continuous Dynamical Systems-A, 2020, 40(6):3327.
[10] FILIPPAS S, HERRERO M A, VELáZQUEZ J J L.Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 2000, 456(2004):2957-2982.
[11] FUJITA H..:On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha}$ . J. Fac. Sci. Univ. Tokyo Sect. I . J. Amer. Math. Soc. 28 (2015),
[12] GIGA Y, KOHN R V. Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 1985, 38(3):297-319.
[13] GIGA Y, KOHN R V. Characterizing blowup using similarity variables. Indiana Univ. Math. J., 1987, 36(1):1-40.
[14] GIGA Y, MATSUI S, SASAYAMA S. Blow up rate for semilinear heat equations with subcritical nonlinearity. Indiana Univ. Math. J., 2004, 53(2):483-514.
[15] HARADA J. Blowup profile for a complex valued semilinear heat equation. J. Funct. Anal., 2016, 270(11):4213-4255.
[16] HARADA J. Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension. Calc. Var. Partial Differential Equations, 2017, 56(4):Paper No. 121, 36.
[17] HARADA J. A higher speed type II blowup for the five dimensional energy critical heat equation.. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2020, 37(2):309-341.
[18] HARADA J. A type ii blowup for the six dimensional energy critical heat equation. arXiv:2002.00528, 2020.
[19] HERRERO M A, VELáZQUEZ J J L. Explosion de solutions d’équations paraboliques semilinéaires supercritiques. C. R. Acad. Sci. Paris Sér. I Math., 1994, 319(2):141-145.
[20] HERRERO M A, VELÁZQUEZ J J. A blow up result for semilinear heat equations in the supercritical case. unpublished, 1992.
[21] MATANO H, MERLE F. On nonexistence of type II blowup for a supercritical nonlinear heat equation. Comm. Pure Appl. Math., 2004, 57(11):1494-1541.
[22] MERLE F, ZAAG H. Stability of the blow-up profile for equations of the type $u_{t}=\Delta u+|u|^{p-1}u$ . Duke Math. J., 1997, 86(1):143-195.
[23] MERLE F, RAPHAëL P, SZEFTEL J. On Strongly Anisotropic Type I Blowup. Int. Math. Res. Not. IMRN, 2020(2):541-606.
[24] POON C C. Blow-up behavior for semilinear heat equations in nonconvex domains[J]. Differential Integral Equations, 2000, 13(7-9):1111-1138.
[25] QUITTNER P, SOUPLET P. Birkhäuser advanced texts: Basler lehrbücher. [birkhäuser advanced texts: Basel textbooks]: Superlinear parabolic problems.. Birkhäuser/Springer, Cham, 2019: xvi+725.
[26] SCHWEYER R. Type II blow-up for the four dimensional energy critical semi linear heat equation. J. Funct. Anal., 2012, 263(12):3922-3983.
[27] SEKI Y. Type II blow-up mechanisms in a semilinear heat equation with critical Joseph-Lundgren exponent. J. Funct. Anal., 2018, 275(12):3380-3456.
[28] TALENTI G. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 1976, 110:353-372.
[29] VELáZQUEZ J J L. Estimates on the $(n-1)$ -dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J., 1993, 42(2):445-476.

Construction of blow-up solution for 5 dimentional critical fujita type equation with different blow-up speed

Abstract.

1. Introduction

1.1. Main result

Theorem 1.1.

Remark 1.2.

2. Basic facts in the construction

2.1. The Construction of the approximated solutions

2.2. Inner-outer gluing

2.3. A result about the linearized equation

Lemma 2.1.

2.4. Local properties of the heat equation

Lemma 2.2.

Lemma 2.3.

Lemma 2.4.

3. The parameters λi\lambda_{i} and ξi\xi_{i}

3.1. Formal derivation of λi\lambda_{i} and ξi\xi_{i}

3.2. The selection of λi\lambda_{i} and ξi\xi_{i}

Lemma 3.1.

4. The inner problem

Proposition 4.1.

5. The outer problem

5.1. Choosing of the parameters

5.2. The estimate of G

Lemma 5.1.

Proposition 5.2.

5.3. The solving of Φ1\Phi_{1}

Lemma 5.3.

5.4. The solving of Φ2\Phi_{2}

Lemma 5.4.

Lemma 5.5.

5.5. A point wise estimate for ψ\psi in |z|>eτ4|z|>e^{\frac{\tau}{4}}

Lemma 5.6.

Lemma 5.7.

6. The proof of theorem 1.1

References

3. The parameters $\lambda_{i}$ and $\xi_{i}$

3.1. Formal derivation of $\lambda_{i}$ and $\xi_{i}$

3.2. The selection of $\lambda_{i}$ and $\xi_{i}$

5.3. The solving of $\Phi_{1}$

5.4. The solving of $\Phi_{2}$

5.5. A point wise estimate for $\psi$ in $|z|>e^{\frac{\tau}{4}}$