Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient

Abstract

Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition are investigated. We derive the conditions on the data of problem (1.1) sufficient to guarantee that blow-up will occur, and obtain an upper bound for $t^{\ast }$. Also we give the condition for global existence of the solution.

1 Introduction

In this work, we consider the following non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition

$$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u+{\int }_{\mathrm{\Omega }}{u}^{p}dx-k(t)u^q,x=(x_1,x_2,\dots ,x_N)\in \mathrm{\Omega },t\in (0,t^{\ast }),\\ u(x,t)=0,x\in \partial \mathrm{\Omega },t\in (0,t^{\ast }),\\ u(x,0)=u_0(x)\ge 0,x\in \mathrm{\Omega },\end{array}$$

(1.1)

where $\mathrm{\Omega }\subset R^N$ is a bounded domain with a smooth boundary ∂ Ω, Δ is the Laplace operator, and $t^{\ast }$ is the possible blow-up time. By the maximum principle, it follows that $u(x,t)\ge 0$ in the time interval of existence. The coefficient $k(t)$ is assumed to be nonnegative. The particular case of $k=\mathrm{const}$ of problem (1.1) has already been investigated by many authors, in [1, 2], they studied the question of blow-up for the solution, and in [3–5], they derived lower bounds for blow-up time under different boundary conditions. To deal with problem (1.1) with time dependent coefficient, we make the assumption on the parameters p and q, that is, $p=q>1$.

The motivation of this article comes from the work of Payne and Philippin in [6], where they investigated the blow-up phenomena of the solution of the following problem

$$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u+k(t)f(u),x=(x_1,\dots ,x_N)\in \mathrm{\Omega },t\in (0,t^{\ast }),\\ u(x,t)=0,x\in \partial \mathrm{\Omega },\\ u(x,0)=u_0(x),x\in \mathrm{\Omega },\end{array}$$

(1.2)

where Ω is a bounded sufficiently smooth domain in $R^N$, $N\ge 2$, and the coefficient $k(t)$ is assumed nonnegative or strictly positive depending on the situation.

In next, we employ a method used by Kaplan in [7] to obtain a condition, which leads to blow-up at some finite time and also leads to an upper bound for the blow-up time. In Section 3, we derive the condition on the data of problem (1.1) sufficient to guarantee the global existence of $u(x,t)$.

2 Conditions for blow-up in finite time $t^{\ast }$

Let ${\lambda }_1$ be the first eigenvalue, and let ${\varphi }_1$ be the associated eigenfunction of the Dirichlet-Laplace operator defined as

$$\mathrm{\Delta }{\varphi }_1=-{\lambda }_1{\varphi }_1,{\varphi }_1>0,x\in \mathrm{\Omega };{\varphi }_1=0,x\in \partial \mathrm{\Omega },$$

(2.1)

$${\int }_{\mathrm{\Omega }}{\varphi }_{1}dx=1.$$

(2.2)

Let the auxiliary function $\eta (t)$

$$\eta (t):={(|\mathrm{\Omega }|-k(t))}^{\frac{1}{q-1}}{\int }_{\mathrm{\Omega }}u{\varphi }_{1}dx$$

(2.3)

defined in $(0,t^{\ast })$, where $u(x,t)$ is the solution of (1.1) and $q>1$.

We assume that for all $t\in (0,t^{\ast })$,

$$|\mathrm{\Omega }|>k(t)>0,\frac{-k^{\prime }(t)}{|\mathrm{\Omega }|-k(t)}\ge \beta ,\underset{x\in \mathrm{\Omega }}{max}{\varphi }_1|\mathrm{\Omega }|\le 1,$$

(2.4)

for some constant β, and

$$\gamma :={\lambda }_1-\frac{\beta }{q-1}.$$

(2.5)

We deduce from (2.4) and (2.5) that

$$\begin{array}{rl}{\eta }^{\prime }(t)& \ge \frac{\beta }{q-1}\eta (t)+{(|\mathrm{\Omega }|-k(t))}^{\frac{1}{q-1}}{\int }_{\mathrm{\Omega }}{\varphi }_1[\mathrm{\Delta }u+{\int }_{\mathrm{\Omega }}{u}^{q}dx-k(t)u^q]dx\\ =-\gamma \eta (t)+{(|\mathrm{\Omega }|-k(t))}^{\frac{1}{q-1}}({\int }_{\mathrm{\Omega }}{u}^{q}dx-k(t){\int }_{\mathrm{\Omega }}{\varphi }_1{u}^{q}dx)\\ \ge -\gamma \eta (t)+{(|\mathrm{\Omega }|-k(t))}^{\frac{1}{q-1}}\left((\frac{1}{{max}_{x\in \mathrm{\Omega }}{\varphi }_1}-k(t)){\int }_{\mathrm{\Omega }}{\varphi }_1{u}^{q}dx\right)\\ \ge -\gamma \eta (t)+{(|\mathrm{\Omega }|-k(t))}^{\frac{q}{q-1}}{\int }_{\mathrm{\Omega }}{\varphi }_1{u}^{q}dx.\end{array}$$

(2.6)

Furthermore, using (2.2) and Hölder’s inequality, we get

$${\int }_{\mathrm{\Omega }}{\varphi }_{1}udx\le {\left({\int }_{\mathrm{\Omega }}{\varphi }_1{u}^{q}dx\right)}^{\frac{1}{q}}.$$

(2.7)

Combining (2.6) and (2.7), we get

$${\eta }^{\prime }(t)\ge -\gamma \eta (t)+{\eta }^q(t),t\in (0,t^{\ast }).$$

(2.8)

By integrating (2.8), we get

$${(\eta (t))}^{1-q}\le \mathrm{\Theta }(t):=\{\begin{array}{ll}({(\eta (0))}^{1-q}-\frac{1}{\gamma })e^{\gamma (q-1)t}+\frac{1}{\gamma },& \gamma \ne 0,\\ {(\eta (0))}^{1-q}+(1-q)t,& \gamma =0.\end{array}$$

(2.9)

If $\mathrm{\Theta }(T_1)=0$ for some $T_1>0$, then $\eta (t)$ blows up at time $t^{\ast }<T_1$. This result is summarized in the following theorem.

Theorem 1 Let $u(x,t)$ be the solution of problem (1.1). Then the auxiliary function $\eta (t)$ defined in (2.3) blows up at time $t^{\ast }<T_1$ with

$$T_1:=\{\begin{array}{ll}\frac{1}{\gamma (q-1)}log(-\frac{1}{\gamma ({(\eta (0))}^{1-q}-\frac{1}{\gamma })})& \mathit{\text{if }}0<\gamma {(\eta (0))}^{1-q}<1,\\ \frac{1}{(q-1){(\eta (0))}^{q-1}}& \mathit{\text{if }}\gamma \le 0.\end{array}$$

(2.10)

3 Condition for global existence

In this section, our argument makes use of the following Sobolev-type inequality

$${\left({\int }_{\mathrm{\Omega }}{v}^{6}dx\right)}^{1/4}\le \mathrm{\Gamma }{({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{3/4},\mathrm{\Gamma }=\frac{{2.3}^{-3/4}}{\pi },$$

(3.1)

valid in $R^3$ for a nonnegative function v that vanishes on ∂ Ω. In this section, our results restricted to $R^3$ for proof of (3.1), see [8].

We consider the auxiliary function $\sigma (t)$ defined as

$$\sigma (t):=M^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}{\int }_{\mathrm{\Omega }}{u}^{2n(p-1)}dx,t\in (0,t^{\ast }),$$

(3.2)

with

$$M:={(|\mathrm{\Omega }|-k(0))}^{2n}{\int }_{\mathrm{\Omega }}{u}_0^{2n(q-1)}dx,$$

(3.3)

we assume that for all $t\in (0,t^{\ast })$,

$$|\mathrm{\Omega }|>k(t)>0,\frac{-k^{\prime }(t)}{|\mathrm{\Omega }|-k(t)}<\beta ,$$

(3.4)

for some constant β. In (3.2)-(3.3), n is subjected to restrictions

$$n(q-1)\ge 1,n>\frac{3}{4}.$$

(3.5)

For convenience, we set

$$v(x,t)=u^{n(q-1)},$$

(3.6)

and compute

$$\begin{array}{rcl}{\sigma }^{\prime }(t)& =& 2n\frac{-k^{\prime }(t)}{|\mathrm{\Omega }|-k(t)}\sigma (t)+2n(q-1)M^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}\\ \times {\int }_{\mathrm{\Omega }}{u}^{2n(q-1)-1}[\mathrm{\Delta }u+{\int }_{\mathrm{\Omega }}{u}^{q}dx-k(t)u^q]dx\end{array}$$

(3.7)

with

$${\int }_{\mathrm{\Omega }}{u}^{2n(q-1)-1}\mathrm{\Delta }udx=-\frac{2n(q-1)-1}{n^2{(q-1)}^2}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx,$$

(3.8)

due to (3.4), we obtain

$$\begin{array}{rcl}{\sigma }^{\prime }(t)& \le & 2n\beta \sigma (t)-\frac{2[2n(q-1)-1]}{n(q-1)}{(|\mathrm{\Omega }|-k(t))}^{2n}{M}^{-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx\\ +2n(q-1)M^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}[|\mathrm{\Omega }|{\int }_{\mathrm{\Omega }}{v}^{2+\frac{1}{n}}dx-k(t){\int }_{\mathrm{\Omega }}{v}^{2+\frac{1}{n}}dx]\\ =& 2n\beta \sigma (t)-\frac{2[2n(q-1)-1]}{n(q-1)}{(|\mathrm{\Omega }|-k(t))}^{2n}{M}^{-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx\\ +2n(q-1)M^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n+1}{\int }_{\mathrm{\Omega }}{v}^{2+\frac{1}{n}}dx.\end{array}$$

(3.9)

By using Hölder’s inequality,

$${\int }_{\mathrm{\Omega }}{v}^{2+\frac{1}{n}}dx\le {\left({\int }_{\mathrm{\Omega }}{v}^{2}dx\right)}^{(4n-1)/4n}{\left({\int }_{\mathrm{\Omega }}{v}^{6}dx\right)}^{1/4n},$$

(3.10)

and Sobolev-type inequality (3.1), we obtain

$$\begin{array}{c}{(|\mathrm{\Omega }|-k(t))}^{2n+1}{\int }_{\mathrm{\Omega }}{v}^{2+\frac{1}{n}}dx\hfill \\ \le {(|\mathrm{\Omega }|-k(t))}^{2n+1}{\left({\int }_{\mathrm{\Omega }}{v}^{2}dx\right)}^{(4n-1)/4n}{({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{3/4n}{\mathrm{\Gamma }}^{1/n}\hfill \\ ={\mathrm{\Gamma }}^{1/n}{M}^{(4n-1)/4n}{\sigma }^{(4n-1)/4n}{({(|\mathrm{\Omega }|-k(t))}^{2n+1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{3/4n},\hfill \end{array}$$

(3.11)

where Γ is defined in (3.1). Joining (3.11) and (3.9), we obtain

$$\begin{array}{c}{\sigma }^{\prime }(t)\hfill \\ \le 2n\beta \sigma (t)-\frac{2[2n(q-1)-1]}{n(q-1)}{M}^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx\hfill \\ +2n(q-1){\mathrm{\Gamma }}^{1/n}{M}^{1/2n}{\sigma }^{(4n-1)/4n}{(M^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{3/4n}\hfill \\ =2n\beta \sigma (t)+2n{({\lambda }^{-1}{M}^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{3/4n}\hfill \\ \times \{{\lambda }^{3/4n}(q-1){\mathrm{\Gamma }}^{1/n}{\sigma }^{(4n-1)/4n}{M}^{1/2n}-\frac{2[2n(q-1)-1]}{n^2(q-1)}\hfill \\ \times \lambda {(M^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}{\lambda }^{-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{(4n-3)/4n}\}\hfill \end{array}$$

(3.12)

with arbitrary $\lambda \ne 0$. Choosing $\lambda :={\lambda }_1$, the first eigenvalue of problem (2.1), we have

$${\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx\ge {\lambda }_1{\int }_{\mathrm{\Omega }}{v}^{2}dx,$$

(3.13)

by the Rayleigh principle. By using (3.13) in the last factor of (3.12), we obtain

$$\begin{array}{rcl}{\sigma }^{\prime }(t)& \le & 2n\beta \sigma (t)+2n{({{\lambda }_1}^{-1}{M}^{-1}{(|\mathrm{\Omega }|-k(t))}^{2n}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{2}dx)}^{3/4n}\\ \times \{{{\lambda }_1}^{3/4n}(q-1){\mathrm{\Gamma }}^{1/n}\sigma {(t)}^{(4n-1)/4n}{M}^{1/2n}-\frac{2[2n(q-1)-1]}{n^2(q-1)}{\lambda }_1\sigma {(t)}^{(4n-3)/4n}\}\\ =& 2n\beta \sigma (t)+2n\sigma {(t)}^{3/4n}\times \sigma {(t)}^{(4n-3)/4n}\{\omega \sigma {(t)}^{1/2n}-(\mu +\beta )\},\end{array}$$

(3.14)

with

$$\omega ={{\lambda }_1}^{3/4n}(q-1){\mathrm{\Gamma }}^{1/n}{M}^{1/2n},\mu =\frac{2[2n(q-1)-1]}{n^2(q-1)}{\lambda }_1-\beta .$$

(3.15)

Suppose that β is small enough to satisfy the condition

$$\mu >0,$$

(3.16)

and that initial data is small enough to satisfy the condition

$$\omega -\mu <0.$$

(3.17)

Then either $\omega {(\sigma (t))}^{1/2n}-\mu$ remains negative for all time, or there exists a first time $t_0$ such that

$$\omega {(\sigma (t_0))}^{1/2n}-\mu =0.$$

(3.18)

Then we obtain the differential inequality

$${\sigma }^{\prime }(t)\le 2n\sigma (t)\{\omega {(\sigma (t))}^{1/2n}-\mu \}\le 0,t\in (0,t_0).$$

(3.19)

Integrating this differential inequality, we obtain

$$\sigma (t)\le {\{(1-\frac{\omega }{\mu })e^{\mu t}+\frac{\omega }{\mu }\}}^{-2n},t>0.$$

(3.20)

This result is summarized in the next theorem.

Theorem 2 Let Ω be a bounded domain in $R^3$, and assume that the data of problem (1.1) satisfy conditions (3.4), (3.16), (3.17). Then the auxiliary function $\sigma (t)$ defined in (3.2) satisfies (3.20), and $u(x,t)$ exists for all time $t>0$.