Global Existence for the Cauchy Problem of the Parabolic–Parabolic–ODE Chemotaxis Model with Indirect Signal Production on the Plane

Abstract

This paper establishes the global existence of solutions to a chemotaxis system with indirect signal production in the whole two-dimensional space. This system exhibits a mass threshold phenomenon governed by a critical mass $m_c=8\pi \delta$, where $\delta$ represents the decay rate of the static individuals. When the total initial mass $m={\int }_{R^2}{u}_{0}dx<m_c$, all solutions exist globally and remain bounded. In the critical case of $m=m_c$, the global existence or finite-time blow-up may occur depending on the initial conditions. The critical mass obtained in the whole space coincides with that previously derived in radially symmetric bounded domains. A key novelty lies in extending the analysis to the full plane, where the absence of compactness is overcome by constructing a suitable Lyapunov functional and employing refined Trudinger–Moser-type inequalities.

1. Introduction

In this paper, we shall consider the Cauchy problem with respect to the following Parabolic–Parabolic–ODE chemotaxis model (1) ($\tau =1$) with indirect signal production

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\Delta u-\nabla ·(u\nabla w),\hfill & \mathrm{in}{R}^2\times (0,\infty ),\hfill \\ \tau w_t=\Delta w-w+v,\hfill & \mathrm{in}{R}^2\times (0,\infty ),\hfill \\ v_t=u-\delta v,\hfill & \mathrm{in}{R}^2\times (0,\infty ),\hfill \\ (u(·,0),w(·,0),v(·,0))=(u_0,w_0,v_0),\hfill & \mathrm{in}{R}^2,\hfill \end{array}\right.\end{array}$$

(1)

where the constant $\delta$ is positive and the initial function $(u_0,w_0,v_0)\in (L^1\left(R^2\right)\cap L^{\infty }\left(R^2\right))\times (L^1\left(R^2\right)\cap H^1\left(R^2\right))\times (L^1\left(R^2\right)\cap L^{\infty }\left(R^2\right))$ is nonnegative. Here $u=u(x,t)$, $w=w(x,t)$ and $v=v(x,t)$ denote the moving individuals’ density, the concentration of chemoattractant, and the static individuals’ density, respectively.

Several studies [1,2,3] have proposed chemotaxis models to characterize the spatial–temporal dynamics of mountain pine beetle populations, which significantly impact North America’s forestry sector. These models quantify the interactions between three key variables: (1) the population density u of flying beetles, (2) the population density v of nesting beetles, and (3) the concentration w of beetle pheromones (chemoattractants). The models distinguish between two behavioral states: flying beetles exhibit random spatial movement with pheromone-mediated directional bias, while nesting beetles remain stationary. Flying beetles u migrate toward regions of higher pheromone concentration w, with the signal being mediated by the nesting beetles v. A critical model feature is that pheromone production originates exclusively from nesting beetles, but it only influences the movement of flying beetles. This mechanistic difference fundamentally distinguishes these models from classical Keller–Segel systems [4,5,6,7], where the chemotactic agent is both produced by and acts upon the same population. As demonstrated in subsequent analysis, this distinction leads to markedly different population dynamics, particularly in two-dimensional space.

The chemotaxis system with indirect signal production has been studied in a bounded domain $\Omega \subset R^2$ under the Neumann boundary condition. For a Parabolic–Parabolic–ODE system with indirect signal production,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\Delta u-\nabla ·(u\nabla w),\hfill & x\in \Omega ,t>0\hfill \\ w_t=\Delta w-w+v,\hfill & x\in \Omega ,t>0,\hfill \\ v_t=u-\delta v,\hfill & x\in \Omega ,t>0,\hfill \\ {\partial }_{\nu }u={\partial }_{\nu }w={\partial }_{\nu }v=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,w(x,0)=w_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0,\hfill & x\in \Omega .\hfill \end{array}\right.\end{array}$$

(2)

If the $w$-equation in (2) is replaced by $0=\Delta w-\mu \left(t\right)+v$, Tao and Winkler [8] obtained a novel critical mass phenomenon and showed that the solution blows up in finite time once m exceeds the critical mass, where $m={\int }_{\Omega }{u}_{0}dx$. In particular, Lautençot [1] showed that any solution of (2) remains bounded for $m\in (0,4\pi \delta )$. Moreover, if $\Omega$ is a ball, then the solutions of (2) exist globally for $m\in (0,8\pi \delta )$.

If $w=u$, Osaki and Yasgi [2] obtained the global existence and convergence behavior of solutions to the 1D Keller–Segel model with generalized sensitivity function by constructing an attractor set, and this result was subsequently improved in [9]. If the $v$-equation in (2) is replaced by $v_t=\Delta v-v+u$, Mao and Li [10] established that, for any prescribed mass $m={\int }_{\Omega }{u}_{0}dx$, there exist radially symmetric positive initial data with the given mass such that the corresponding solutions exhibit finite-time blow-up. Nagai [11] revealed that nonradial solutions developed singularities within finite time, provided the initial data’s mass and second moment obey certain quantitative conditions. The enhancements can be attributed to the powerful parabolic smoothing effect resulting from indirect attractant production. Similar beneficial effects on solution boundedness were noted by Chen et al. [12,13] in certain indirect chemotaxis–aptotaxis and rotation models. [14,15] have investigated taxis-driven aggregation phenomena, particularly focusing on the spontaneous formation of high-density regions during intermediate transitions induced by substantial phenotype switching rates. Considering the singularity formation, Tao and Winkler [16] studied a J$\ddot{\mathrm{c}}$ger–Luckhaus-type model, and they proved finite-time blow-up for radial solutions in dimensions $n\ge 5$.

In the whole space $\Omega =R^2$, Calvez [17] established the global existence of solutions of (1) with $w=u$ under the conditions that $m(u_0,R^2)<8\pi$, $u_{0}log{u}_0\in L^1\left(R^2\right)$ and $u_0{log(1+|x|}^2)\in L^1\left(R^2\right)$. In [18], the analysis relaxed previous auxiliary conditions, and they obtained a global existence solution if $m(u_0,R^2)<4\pi$. This critical threshold arises naturally from the underlying Brezis–Merle inequality in the proof framework. Subsequently, studies in [19,20] showed that solutions with $m(u_0;R^2)>8\pi$ exhibit finite-time blow-up. Furthermore, Mizoguchi [21] employed the Trudinger–Moser inequality to establish the global existence of solutions without additional conditions. Furthermore, this analysis was extended to the critical case where $m(u_0;R^2)=8\pi$ under suitable regularity conditions on the initial data $u_0$, thereby generalizing the earlier results of Calvez [17]. Xiang and Yang [22] considered parabolic–elliptic–ODE systems with indirect signal production, in the radially symmetric setting, which obtained uniform boundedness for $m<8\pi \delta$, while the solutions may blow up with $m>8\pi \delta$.

In [1], by constructing the Lyapunov functional

$$\begin{array}{c}\hfill \mathcal{F}(u,w,v)={\int }_{\Omega }ulnudx-{\int }_{\Omega }uwdx+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{w}_t^{2}dx+{\displaystyle \frac{\delta +1}{2}}{\int }_{\Omega }{|\nabla w|}^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{\Omega }{w}^{2}dx,\end{array}$$

(3)

and employing the Trudinger–Moser inequality, Laurençot deduced the critical mass $m_c=8\pi \delta$ that sharply separates solutions exhibiting global boundedness from those undergoing finite-time blow up. Similarly to [1], we will obtain the global existence of solutions in a two-dimensional setting by constructing the Lyapunov functional

$$\begin{array}{cc}\hfill \mathcal{F}(u,w,v)& ={\int }_{R^2}(u+1)log(u+1)dx-{\int }_{R^2}uwdx+{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx\hfill \\ & +{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{|\nabla w|}^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx.\hfill \end{array}$$

(4)

We note that the functional $F(u,w,v)$ is not actually independent of v, and the v-terms are implicitly expressed through w due to the relationship established by the w-equation. Specifically, v has been replaced or reformulated in terms of w to simplify the expression or to highlight a particular structure in the problem.

Here, we choose $(u+1)log(u+1)$ over $ulogu$ to simplify the estimates: for $u<1$, the latter become negative and would require extra handling in the inequalities, whereas $(u+1)log(u+1)$ is always non-negative and keeps the desired coefficient positive.

Our proof employs the Trudinger–Moser inequality, which inherently requires domains of finite Lebesgue measure and thus cannot be directly applied to $R^2$. We address this limitation through an innovative approach, which is analogous to Mizoguchi [21]. Building on prior results on [22], we note that the solutions globally exist in time, with $m(u_0;R^2)<4\pi \delta$, and that in the radially symmetric setting, the results could be improved to $m(u_0;R^2)<8\pi \delta$. Now, we can conclude that global existence is guaranteed for all solutions with $m(u_0;R^2)<8\pi \delta$, and finite-time blow-up occurs for certain solutions with $m(u_0;R^2)=8\pi \delta$.

Here, to simplify the notation, we denote ${\rm Y}=(L^1\left(R^2\right)\cap L^{\infty }\left(R^2\right))\times (L^1\left(R^2\right)\cap H^1\left(R^2\right))\times (L^1\left(R^2\right)\cap L^{\infty }\left(R^2\right))$ throughout the paper. Throughout this paper, we consider weak solutions to the system (1), that is, functions $(u,w,v)$ satisfying the equations in the sense of integral identities, with appropriate regularity and integrability conditions specified in the function spaces $L^1,L^{\infty },H^1$. Now, we can state the main theorems as follows:

Theorem 1.

Let $(u_0,w_0,v_0)\in {\rm Y}$. If $m(u_0;R^2)<8\pi \delta$; then the solution of (1) exists globally in time.

For a solution $(u,w,v)$ developing a singularity at time T, a spatial point $a\in R^2$ is designated as a blow-up point if $u(x,t)$ fails to remain locally bounded when $|x-a|\to 0$. The collection of all such points constitutes the blow-up set. The solution exhibits complete blow-up when its blow-up set coincides with the entire plane $R^2$.

Theorem 2.

Let $(u_0,w_0,v_0)\in {\rm Y}$. If $m(u_0;R^2)=8\pi \delta$, then the solution of (1) either exists globally or exhibits finite-time blow-up in $R^2$.

Section 2 establishes Theorem 1 through preliminary lemmas, employing the Trudinger–Moser inequality in $R^2$ as the key tool. The critical case $m(u_0;R^2)=8\pi \delta$ is addressed in Section 3, where we develop refined arguments building on Theorem 1’s proof to demonstrate Theorem 2.

2. Proof of Theorem 1

For a solution $(u,w,v)$ of (1), we define

$$\begin{array}{cc}\hfill E\left(t\right)& ={\int }_{R^2}\left(u+1\right)log\left(u+1\right)dx-{\int }_{R^2}uwdx+{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx\hfill \\ \hfill & +{\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \end{array}$$

(5)

for all $t\in (0,T)$, where T is the maximum existence time.

As shown in [18] (Proposition 4.1), the first result requires no additional proof, while the second emerges as a variation on the established Cauchy–Neumann problem in the bounded domain.

Proposition 1.

Let $(u_0,w_0,v_0)\in {\rm Y}$. Let $(u,w,v)$ be a solution to (1) defined on $[0,T)$ with $T\le \infty$. For all $t\in (0,T)$, the following holds:

(i) ${\left|\right|u\left(t\right)\left|\right|}_1=\left|\right|u_0{\left|\right|}_1$ and

$$\begin{array}{c}\hfill {\left|\right|w\left(t\right)\left|\right|}_1=e^{-t}\left|\right|w_0{\left|\right|}_1+\left(1-e^{-t}\right)\left|\right|v_0{\left|\right|}_1;\end{array}$$

(6)

(ii) Let $E\left(t\right)$ be defined in (5). Then,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}E\left(t\right)+\left|\right|w_t{\left|\right|}_2^2+{\int }_{R^2}u\left(t\right){|\nabla \left(log\left(u\right(t)+1)-w\left(t\right)\right)|}^{2}dx\hfill \\ & +{\int }_{R^2}{|\nabla w_t|}^{2}dx+{\int }_{R^2}|\nabla \left[log(u\left(t\right)+1)-{\displaystyle \frac{1}{2}}w\left(t\right)\right]{|}^{2}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}{\left|\right|\nabla w\left(t\right)\left|\right|}_2^2.\hfill \end{array}$$

(7)

Let $\left|D\right|$ be the Lebesgue measure of $D\subset R^2$. By [23,24]; we recall the following inequality.

Proposition 2.

Let $D\subset R^2$ be a finite-measure domain. There exists a constant $C>0$, independent of D, such that for $g\in H_0^1\left(D\right)$, we have

$$\begin{array}{c}\hfill {\int }_D{e}^{\left|g\right|}dx\le C·\left|D\right|·exp\left({\displaystyle \frac{1}{16\pi }}{\left|\right|\nabla g\left|\right|}_{L^2\left(D\right)}^2\right),\end{array}$$

(8)

namely,

$$\begin{array}{c}\hfill log{\int }_D{e}^{\left|g\right|}dx\le log(C·|D\left|\right)+{\displaystyle \frac{1}{16\pi }}{\left|\right|\nabla g\left|\right|}_{L^2\left(D\right)}^2.\end{array}$$

(9)

The following was proved in [25] Lemma 5.4.

Proposition 3.

Let $D\subset R^2$. For nonnegative integrable functions f and h, we obtain

$$\begin{array}{c}\hfill {\int }_{D}fhdx\le {\int }_{D}flogfdx+Mlog\left({\int }_D{e}^{h}dx\right)-MlogM,\end{array}$$

(10)

where $M={\int }_{D}fdx$.

Lemma 1.

Let $(u_0,w_0,v_0)\in {\rm Y}$. Let $(u,w,v)$ be a solution to (1) defined on $[0,T)$ with $T\le \infty$. For initial data satisfying $\left|\right|u_0{\left|\right|}_1<8\pi \delta$, there exist positive constants $C=C\left(\right|\left|u_0\right|{|}_1,\left|\right|w_0|{|}_1,|\left|v_0\right|{|}_1,T)$ and $\sigma =\sigma \left(\right|\left|u_0\right|{|}_1)>0$ such that

$$\begin{array}{c}\hfill E\left(t\right)\ge \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{w}_t^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{\Omega }{w}^{2}dx-C\end{array}$$

(11)

for all $t\in (0,T)$.

Proof. 

Due to $\left|\right|u_0{\left|\right|}_1<8\pi \delta$, we can select parameters $\sigma$ that are sufficiently small and $s>0$ that are sufficiently large to ensure that

$$\begin{array}{c}\hfill {\displaystyle \frac{\delta +1}{2}}-{\displaystyle \frac{\delta +1}{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\right||w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1)\right\}>0.\end{array}$$

(12)

Let

$$\begin{array}{c}\hfill \tilde{w}(x,t)=max\left\{w(x,t)-s,0\right\}\mathrm{for}(x,t)\in R^2\times (0,T)\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \Sigma \left(t\right)=\{x\in R^2:w(x,t)>s\}\mathrm{for}t\in (0,T).\end{array}$$

(14)

By applying Proposition 1(i), we obtain

$$\begin{array}{c}\hfill {\left|\right|w\left(t\right)\left|\right|}_1\le \left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\mathrm{for}t\in (0,T).\end{array}$$

(15)

Therefore, we infer

$$\begin{array}{c}\hfill s·|\Sigma \left(t\right)|\le {\left|\right|w\left(t\right)\left|\right|}_1\le \left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\mathrm{for}t\in (0,T)\end{array}$$

(16)

and hence

$$\begin{array}{c}\hfill |\Sigma \left(t\right)|<{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\mathrm{for}t\in (0,T).\end{array}$$

(17)

from Proposition 1(i), we deduce that

$$\begin{array}{cc}\hfill {\int }_{R^2}uwdx& \le {\int }_{\Sigma \left(t\right)}u\left(\tilde{w}+s\right)dx+{\int }_{R^2\setminus \Sigma \left(t\right)}uwdx\hfill \\ \hfill & \le {\int }_{\Sigma \left(t\right)}u\tilde{w}dx+s\left|\right|u_0{\left|\right|}_1.\hfill \end{array}$$

(18)

If $\Sigma \left(t\right)=Ø$, then the conclusion of this lemma follows immediately. For $t\in (0,T)$ with $\Sigma \left(t\right)\ne Ø$, the set $\Sigma \left(t\right)$ can be decomposed into at most countably many pairwise disjoint domains $\left\{{\Sigma }_j\left(t\right):j=1,2,\dots \right\}$, that is, $\Sigma \left(t\right)={\bigcup }_{j=1}^{m\left(t\right)}{\Sigma }_j\left(t\right)$ with $m\left(t\right)\in \mathbb{N}\cup \infty$, where ${\Sigma }_j\left(t\right)$ is a connected and open domain for $j=1,2,\dots ,m\left(t\right)$ and ${\Sigma }_k\left(t\right)\cap {\Sigma }_j\left(t\right)=Ø$ if $k\ne j$. Letting $f=\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(1+u)$ and $h=\sqrt{{\displaystyle \frac{\delta +1}{\delta }}}w$ in Proposition 3, we deduce

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }_j\left(t\right)}(u+1)\tilde{w}dx& \le {\int }_{{\Sigma }_j\left(t\right)}(1-\sigma )\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\left[(1-\sigma )\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)\right]dx\hfill \\ \hfill & +(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)log({\int }_{{\Sigma }_j\left(t\right)}{e}^{{\displaystyle \frac{\sqrt{\frac{(1+\delta )}{\delta }}\tilde{w}}{1-\sigma }}}dx)\hfill \\ \hfill & -(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)log\left[(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\right]\hfill \end{array}$$

(19)

for $j=1,2,\dots ,m\left(t\right)$, where $M_j\left(t\right)={\int }_{{\Sigma }_j\left(t\right)}u\left(t\right)dx$. From Proposition 2, we infer

$$\begin{array}{c}\hfill log{\int }_{{\Sigma }_j\left(t\right)}{e}^{{\displaystyle \frac{\sqrt{\frac{\delta +1}{\delta }}\tilde{w}}{1-\sigma }}}dx\le log(c·|{\Sigma }_j\left(t\right)\left|\right)+{\displaystyle \frac{\delta +1}{16\delta \pi {(1-\sigma )}^2}}\left|\right|\nabla \tilde{w}{\left|\right|}_{L^2\left({\Sigma }_j\left(t\right)\right)}^2.\end{array}$$

(20)

Hence, we conclude that

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }_j\left(t\right)}(u+1)\tilde{w}dx& \le (1-\sigma ){\int }_{{\Sigma }_j\left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\left[(1-\sigma )\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)\right]dx\hfill \\ \hfill & +(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\left\{log(c·|{\Sigma }_j\left(t\right)\left|\right)+{\displaystyle \frac{1+\delta }{16\delta \pi {(1-\sigma )}^2}}\left|\right|\nabla \tilde{w}{\left|\right|}_{L^2\left({\Sigma }_j\left(t\right)\right)}^2\right\}\hfill \\ \hfill & -(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)log\left[(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\right]\hfill \\ \hfill & \le (1-\sigma )\left\{{\int }_{{\Sigma }_j\left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx+log(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\right\}\hfill \\ \hfill & +(1-\sigma )\{\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)log\left|{\Sigma }_j\left(t\right)\right|+logc·\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi {(1-\sigma )}^2}}\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right){\int }_{{\Sigma }_j\left(t\right)}{|\nabla \tilde{w}|}^{2}dx\}\hfill \\ \hfill & -(1-\sigma )\left\{\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)log(1-\sigma )+\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)log\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\right\}\hfill \\ \hfill & \le (1-\sigma ){\int }_{{\Sigma }_j\left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right){\int }_{{\Sigma }_j\left(t\right)}{|\nabla \tilde{w}|}^{2}dx\hfill \\ & +(1-\sigma )\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)logc\hfill \end{array}$$

(21)

for $j=1,2,\dots ,m\left(t\right)$ since

$$\begin{array}{c}\hfill log\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right)\ge log\left|{\Sigma }_j\left(t\right)\right|.\end{array}$$

(22)

Let $M\left(t\right)={\int }_{\Sigma \left(t\right)}u\left(t\right)dx$. It can be inferred that

$$\begin{array}{cc}\hfill {\int }_{\Sigma \left(t\right)}(u+1)\tilde{w}dx& =\sum _{j=1}^{m\left(t\right)}{\int }_{{\Sigma }_j\left(t\right)}(u+1)\tilde{w}dx\hfill \\ \hfill & \le (1-\sigma ){\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left(M_j\left(t\right)+\left|{\Sigma }_j\left(t\right)\right|\right){\int }_{\Sigma \left(t\right)}{|\nabla \tilde{w}|}^{2}dx+(1-\sigma )\left(M\left(t\right)+|\Sigma (t\left)\right|\right)·logc\hfill \end{array}$$

(23)

since

$$\begin{array}{c}\hfill {\int }_{{\Sigma }_j\left(t\right)}|\nabla \tilde{w}{|}^{2}dx\le {\int }_{\Sigma \left(t\right)}{|\nabla \tilde{w}|}^{2}dx.\end{array}$$

(24)

Making use of $M\left(t\right)\le \left|\right|u_0{\left|\right|}_1$ and (17), we obtain

$$\begin{array}{cc}\hfill {\int }_{\Sigma \left(t\right)}(u+1)\tilde{w}dx& \le (1-\sigma ){\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(25)

Hence, we infer from the above inequalities that

$$\begin{array}{cc}\hfill {\int }_{R^2}uwdx& \le (1-\sigma ){\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(26)

Consequently, we deduce

$$\begin{array}{cc}\hfill E\left(t\right)& =\sigma {\int }_{R^2}\left(u(x,t)+1\right)log\left(u(x,t)+1\right)dx\hfill \\ & +(1-\sigma )({\int }_{R^2}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u(x,t)+1\right)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u(x,t)+1\right)dx\hfill \\ & -{\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx)\hfill \\ \hfill & +{\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx-{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ & +{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx-(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(27)

By the choice of $\sigma$, s, we have

$$\begin{array}{cc}\hfill E\left(t\right)& \ge \sigma {\int }_{R^2}\left(u(x,t)+1\right)log\left(u(x,t)+1\right)dx+{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \\ \hfill & -(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(28)

This completes the proof.  □

Lemma 2.

Let $(u_0,w_0,v_0)\in {\rm Y}$. Let $(u,w,v)$ be a solution to (1) defined on $[0,T)$ with $T\le \infty$. If $\left|\right|u_0{\left|\right|}_1<8\pi \delta$, then there exist constants $C=C\left(\right|\left|u_0\right|{|}_1,\left|\right|w_0|{|}_1,|\left|v_0\right|{|}_1,T)$, $\sigma =\sigma \left(\right|\left|u_0\right|{|}_1)>0$ such that

$$\begin{array}{c}\hfill \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\int }_0^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_0^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau \le C\end{array}$$

(29)

for $t\in (0,T)$.

Proof. 

Similarly, there exists a $C=C\left(\right|\left|u_0\right|{|}_1,\left|\right|w_0|{|}_1,|\left|v_0\right|{|}_1,T)>0$ such that

$$\begin{array}{cc}\hfill {\int }_{R^2}uwdx& \le {\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(30)

for $t\in (0,T)$. This entails

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx& =E\left(t\right)-{\int }_{R^2}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)dx+{\int }_{R^2}uwdx\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx-{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \\ \hfill & \le E\left(t\right)-{\int }_{R^2}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)dx+{\int }_{\Sigma \left(t\right)}(u+1)log(u+1)dx\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx-{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \end{array}$$

(31)

for all $t\in (0,T)$. Therefore, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx& \le E\left(t\right)+{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc\hfill \end{array}$$

(32)

for $t\in (0,T)$. Set

$$\begin{array}{c}\hfill \iota ={\displaystyle \frac{\delta +1}{2}}-{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1+{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}.\end{array}$$

(33)

For sufficiently large s, then $\iota >0$ since $\left|\right|u_0{\left|\right|}_1<8\pi \delta$. It follows from (57) that

$$\begin{array}{c}\hfill {\int }_{R^2}{|\nabla w|}^{2}dx\le {\displaystyle \frac{1}{\iota }}\left\{E\left(t\right)+C\right\}\mathrm{for}t\in (0,T),\end{array}$$

(34)

which entails that

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}E\left(t\right)+\left|\right|\nabla w_t{\left|\right|}_2^2+\left|\right|w_t{\left|\right|}_2^2\le {\displaystyle \frac{1}{4\iota }}\left\{E\left(t\right)+C\right\}\mathrm{for}t\in (0,T),\end{array}$$

(35)

by Proposition 1(ii). Therefore, we infer

$$\begin{array}{c}\hfill E\left(t\right)+{\int }_0^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_0^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau +C\le \left\{E\left(0\right)+C\right\}{e}^{{\displaystyle \frac{t}{4\iota }}}\mathrm{for}t\in (0,T).\end{array}$$

(36)

Consequently there exist some $\tilde{C}=\tilde{C}\left(\right||u_0{\left|\right|}_1,\left|\right|w_0{\left|\right|}_1,\left|\right|v_0{\left|\right|}_1,T)$, $\sigma =\sigma \left(\right|\left|u_0\right|{|}_1)>0$ such that

$$\begin{array}{c}\hfill \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\int }_0^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_0^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau \le \tilde{C}\mathrm{for}t\in (0,T).\end{array}$$

(37)

We can complete the proof. □

The following holds in [18] Proposition 5.1.

Proposition 4.

Let $(u_0,w_0,v_0)\in {\rm Y}$. Let $(u,w,v)$ be any solution to (1) defined on $[0,T)$ with $T\le \infty$. Assume that there are positive constants C and σ such that

$$\begin{array}{c}\hfill \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\int }_0^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_0^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau \le C\end{array}$$

(38)

for all $t\in (0,T)$; thus, there exists a positive constant $C_1$ such that

$$\begin{array}{c}\hfill {\left|\right|u\left(t\right)\left|\right|}_2\le C_1\mathrm{for}t\in (0,T].\end{array}$$

(39)

Now we start to prove Theorem 1.

Proof of Theorem 1. 

In contrast, there is a solution $(u,w,v)$ of (1) satisfying the hypotheses that blows up in finite time T. From Lemma 2, we have positive constants $C_1,\sigma$ with

$$\begin{array}{c}\hfill \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\int }_0^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_0^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau \le C_1\end{array}$$

(40)

for $t\in (0,T)$. Consequently, by applying Proposition 4, we obtain $C_2>0$, satisfying

$$\begin{array}{c}\hfill {\left|\right|u\left(t\right)\left|\right|}_2\le C_2\mathrm{for}t\in (0,T].\end{array}$$

(41)

□

Using standard parabolic regularity arguments shows $(u,w,v)$ remains bounded at $t=T$, contradicting our assumption.

3. The Critical Case m(u₀; R²) = 8πδ

Lemma 3.

Let $(u_0,w_0,v_0)\in {\rm Y}$ fulfill $\left|\right|u_0{\left|\right|}_1=8\pi \delta$. Consider a solution $(u,w,v)$ of (1) with blow-up time $T<+\infty$. If the blow-up set of $(u,w,v)$ is a proper subset of $R^2$, then there exist positive constants $C=C\left(\right|\left|u_0\right|{|}_1,\left|\right|w_0|{|}_1,|\left|v_0\right|{|}_1,T)$ and $\sigma =\sigma \left(\right|\left|u_0\right|{|}_1,\left|\right|w_0\left|{|}_{L^1}\right)>0$ for which

$$\begin{array}{c}\hfill E\left(t\right)\ge \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx-C\end{array}$$

(42)

where $t\in (T/2,T)$.

Proof. 

Since $(u,w,v)$ does not blow up in the plane, there exists a point $y_0\in R^2$ and $c_0,R_0>0$ such that

$$\begin{array}{c}\hfill 0\le u(x,t)\le c_0\mathrm{in}{B}_{R_0}\left(y_0\right)\times [0,T),\end{array}$$

(43)

where $B_r\left(b\right)$ denotes a ball with radius r centered as b for $b\in R^2$ and $r>0$. Using the parabolic regularity theory, there exists a constant $R_1>0$ such that $u(x,T)\equiv {lim}_{t\to T}u(x,t)$ exists for $y\in B_{R_1}\left(y_0\right)$. Since u is positive in $R^2\times (0,T)$, we conclude that $u(y,T)>0$ for $y\in B_{R_1}\left(y_0\right)$ by the maximum principle. Hence there exist $b_0,R_2>0$ such that

$$\begin{array}{c}\hfill u(y,t)\ge b_0\mathrm{in}{B}_{R_2}\left(y_0\right)\times [T/2,T).\end{array}$$

(44)

We can choose $\sigma ,s>0$ that are sufficiently small and large such that

$$\begin{array}{c}\hfill {\displaystyle \frac{\delta +1}{2}}-{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·|B_{R_2}\left(x_0\right)|+{\displaystyle \frac{1+b_0}{s}}\left(\right||w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1)\right\}>0,\end{array}$$

(45)

Similar to the steps (13)–(17) in Lemma 1, we use (44) and (17) to infer that

$$\begin{array}{c}\hfill {\int }_{B_{R_2}\left(y_0\right)\setminus \Sigma \left(t\right)}u(y,t)dy>b_0\left\{|B_{R_2}\left(y_0\right)|-{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}\end{array}$$

(46)

for $t\in [T/2,T)$. Setting $M\left(t\right)={\int }_{\Sigma \left(t\right)}u\left(t\right)dx$, we get

$$\begin{array}{cc}\hfill M\left(t\right)& ={\int }_{R^2}u(y,t)dy-{\int }_{R^2\setminus \Sigma \left(t\right)}u(y,t)dy\hfill \\ \hfill & \le \left|\right|u_0{\left|\right|}_1-{\int }_{B_{R_2\left(y_0\right)}\setminus \Sigma \left(t\right)}u(y,t)dy\hfill \\ \hfill & <\left|\right|u_0{\left|\right|}_1-b_0\left\{|B_{R_2}\left(y_0\right)|-{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\},\hfill \end{array}$$

(47)

namely,

$$\begin{array}{c}\hfill M\left(t\right)<\left|\right|u_0{\left|\right|}_1-b_0\left\{|B_{R_2}\left(y_0\right)|-{\displaystyle \frac{1}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\},\end{array}$$

(48)

for $t\in [T/2,T)$. Let

$$\begin{array}{c}\hfill \tilde{w}(x,t)=max\left\{w(y,t)-s,0\right\}\mathrm{for}(y,t)\in R^2\times [T/2,T).\end{array}$$

(49)

From Proposition 1(i), we deduce

$$\begin{array}{cc}\hfill {\int }_{R^2}uwdx& \le {\int }_{\Sigma \left(t\right)}u\left(\tilde{w}+s\right)dx+{\int }_{R^2\setminus \Sigma \left(t\right)}uwdx\hfill \\ \hfill & \le {\int }_{\Sigma \left(t\right)}u\tilde{w}dx+s\left|\right|u_0{\left|\right|}_1.\hfill \end{array}$$

(50)

If $\Sigma \left(t\right)=Ø$, then the conclusion of this lemma follows immediately. For $t\in [T/2,T]$ for which $\Sigma \left(t\right)\ne Ø$, arguing as in (23), we derive that

$$\begin{array}{cc}\hfill {\int }_{\Sigma \left(t\right)}(u+1)\tilde{w}dx& \le (1-\sigma ){\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)dx\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left(M\left(t\right)+|\Sigma (t\left)\right|\right){\int }_{\Sigma \left(t\right)}{|\nabla \tilde{w}|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left(M\left(t\right)+|\Sigma (t\left)\right|\right)logc.\hfill \end{array}$$

(51)

Hence we infer from the above inequalities that

$$\begin{array}{cc}\hfill {\int }_{R^2}uwdx& \le (1-\sigma ){\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(52)

Hence this implies

$$\begin{array}{cc}\hfill E\left(t\right)& =\sigma {\int }_{R^2}\left(u(x,t)+1\right)log\left(u(x,t)+1\right)dx\hfill \\ & +(1-\sigma )\left({\int }_{R^2}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u(x,t)+1\right)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u(x,t)+1\right)dx-{\int }_{\Sigma \left(t\right)}(u+1)log(u+1)dx\right)\hfill \\ \hfill & +{\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx-{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}\hfill \\ & {\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx+{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \\ & -(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(53)

This completes the proof.  □

Lemma 4.

Let $(u_0,w_0,v_0)\in {\rm Y}$ fulfill $\left|\right|u_0{\left|\right|}_1=8\pi \delta$. Consider a solution $(u,w,v)$ of (1) with blow-up time $T<+\infty$. If the blow-up set of $(u,w,v)$ is a proper subset of $R^2$, then there exist positive constants $C=C\left(\right|\left|u_0\right|{|}_1,\left|\right|w_0|{|}_1,|\left|v_0\right|{|}_1,T)$ and $\sigma =\sigma \left(\right|\left|u_0\right|{|}_1,\left|\right|w_0\left|{|}_1\right)>0$ for which

$$\begin{array}{c}\hfill \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\int }_{T/2}^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_{T/2}^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau \le C\end{array}$$

(54)

for $t\in [T/2,T)$.

Proof. 

Similarly, there exists a $C=C\left(\right|\left|u_0\right|{|}_1,\left|\right|w_0|{|}_1,|\left|v_0\right|{|}_1,T)>0$ such that

$$\begin{array}{cc}\hfill {\int }_{R^2}uwdx& \le {\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc.\hfill \end{array}$$

(55)

for $t\in [T/2,T)$. This entails

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx& =E\left(t\right)-{\int }_{R^2}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}\left(u+1\right)dx+{\int }_{R^2}uwdx\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx-{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \\ \hfill & \le E\left(t\right)-{\int }_{R^2}\left(u+1\right)log\left(u+1\right)dx+{\int }_{\Sigma \left(t\right)}\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)log\sqrt{{\displaystyle \frac{\delta }{1+\delta }}}(u+1)dx\hfill \\ \hfill & +{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}{\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx\hfill \\ \hfill & +(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_{R^2}{w}_t^{2}dx-{\displaystyle \frac{\delta }{2}}{\int }_{R^2}{w}^{2}dx\hfill \end{array}$$

(56)

for all $t\in [T/2,T)$. Therefore, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta +1}{2}}{\int }_{R^2}{|\nabla w|}^{2}dx& \le E\left(t\right)+{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}\hfill \\ \hfill & {\int }_{\Sigma \left(t\right)}{|\nabla w|}^{2}dx+(1-\sigma )\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}·logc\hfill \end{array}$$

(57)

for $t\in [T/2,T)$. Set

$$\begin{array}{c}\hfill \iota ={\displaystyle \frac{\delta +1}{2}}-{\displaystyle \frac{1+\delta }{16\delta \pi (1-\sigma )}}\left\{\left|\right|u_0{\left|\right|}_1-b_0·\left|B_{R_2}\left(y_0\right)\right|+{\displaystyle \frac{1+b_0}{s}}\left(\left|\right|w_0{\left|\right|}_1+T\left|\right|v_0{\left|\right|}_1\right)\right\}.\end{array}$$

(58)

For sufficiently large s, then $\iota >0$ since $\left|\right|u_0{\left|\right|}_1=8\pi \delta$. It follows from (57) that

$$\begin{array}{c}\hfill {\int }_{R^2}{|\nabla w|}^{2}dx\le {\displaystyle \frac{1}{\iota }}\left\{E\left(t\right)+C\right\}\mathrm{for}t\in [T/2,T),\end{array}$$

(59)

and thus

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}E\left(t\right)+\left|\right|\nabla w_t{\left|\right|}_2^2+\left|\right|w_t{\left|\right|}_2^2\le {\displaystyle \frac{1}{4\iota }}\left\{E\left(t\right)+C\right\}\mathrm{for}t\in [T/2,T)\end{array}$$

(60)

by Proposition 1(ii). Therefore we infer

$$\begin{array}{c}\hfill E\left(t\right)+{\int }_{T/2}^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_{T/2}^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau +C\le \left\{E\left({\displaystyle \frac{T}{2}}\right)+C\right\}{e}^{{\displaystyle \frac{t}{4\iota }}\left(t-{\displaystyle \frac{T}{2}}\right)}\mathrm{for}t\in [T/2,T).\end{array}$$

(61)

We can complete the proof.  □

Proof of Theorem 2. 

Suppose, for contradiction, that $(u,w,v)$ as a solution of (1) blows up at a finite time T. Then, Lemma 4 provides positive constants $C_1,\sigma$ with

$$\begin{array}{c}\hfill \sigma {\int }_{R^2}(u+1)log(u+1)dx+{\int }_{T/2}^t\left|\right|\nabla w_t{\left(\tau \right)\left|\right|}_2^{2}d\tau +{\int }_{T/2}^t\left|\right|w_t\left(\tau \right){\left|\right|}_2^{2}d\tau \le C_1\end{array}$$

(62)

for all $t\in [T/2,T)$. Then, by applying Proposition 4, we obtain the positive constant $C_2$ satisfying

$$\begin{array}{c}\hfill {\left|\right|u\left(t\right)\left|\right|}_2\le C_2\mathrm{for}t\in [T/2,T].\end{array}$$

(63)

□

Using standard parabolic regularity arguments, we conclude that $(u,w,v)$ remains bounded at $t=T$, contradicting our assumption. We complete the proof of Theorem 2.