Lower bounds for the blow-up time of the nonlinear non-local reaction diffusion problems in ${\mathbb{R}}^N$ ($N\ge 3$)

Abstract

This paper deals with the blow-up of the solution to a non-local reaction diffusion problem in ${\mathbb{R}}^N$ for $N\ge 3$ under nonlinear boundary conditions. Utilizing the technique of a differential inequality, lower bounds for the blow-up time are derived when the blow-up does occur under some suitable assumptions.

MSC: 35K20, 35K55, 35K65.

1 Introduction

There is a vast literature on the question of blow-up of solutions to nonlinear parabolic equations and systems. Readers can refer to the books of Straughan [1] and Quittner and Souple [2], as well as to the survey paper of Bandle and Brunner [3]. For more recent work, one can refer to [4]–[12].

In practical situations, one would like to know among other things whether the solution blows up. In this paper, we consider the blow-up for the solution of the following nonlinear non-local reaction diffusion problems, which have been studied by Song in [4]:

$$\frac{\partial u}{\partial t}=\mathrm{△}u+{\int }_{\mathrm{\Omega }}{u}^{p}dx-k{u}^q\text{in }\mathrm{\Omega }\times (0,t^{\ast }),$$

(1.1)

$$u=0\text{in }\partial \mathrm{\Omega }\times (0,t^{\ast }),$$

(1.2)

$$u(x,0)=f(x)\ge 0\text{in }\mathrm{\Omega },$$

(1.3)

where △ is the Laplace operator, ∂ Ω the boundary of Ω and $t^{\ast }$ the possible blow-up time, $p,q>1$. In [4], [13]–[16], the authors have studied the question of blow-up for the solution of parabolic problems by imposing two different nonlinear boundary conditions: homogeneous Dirichlet boundary conditions or homogeneous Neumann boundary conditions. They determine, for solutions that blow up, a lower bound for the blow-up time $t^{\ast }$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ for $N=3$. Besides, some authors have also started to consider the blow-up phenomena of those problems under Robin boundary conditions (see [17]–[19]). However, for the case $N\ge 3$, the Sobolev inequality, which is important for the result obtained in [4], is no longer applicable. Recently, some papers begin to pay attentions to the study of the blow-up phenomena of solution to an equation in $\mathrm{\Omega }\subset {\mathbb{R}}^N$, for $N\ge 3$ (see [20]–[22]).

In the present paper, for convenience, we set $p=s+1$, $s>0$ and rewrite (1.1) as follows:

$$\frac{\partial u}{\partial t}=\mathrm{△}u+{\int }_{\mathrm{\Omega }}{u}^{s+1}dx-k{u}^q\text{in }\mathrm{\Omega }\times (0,t^{\ast }).$$

(1.4)

As indicated in [23], it is well known that if $p\le q$ the solution will not blow up in finite time. Also it is well known that if the initial data are small enough the solution will actually decay exponentially as $t⟶\mathrm{\infty }$ (see e.g.[1], [24]). Since we are interested in a lower bound for t, in the case of blow-up, we are only concerned with the case $q<p$.

We see by the parabolic maximum principles [25], [26] that u is nonnegative in x for $t\in [0,t^{\ast })$.

In Section 2, we derive the lower bound for the blow-up time of the system (1.1)-(1.3) in ${\mathbb{R}}^N$. The obtained results extend the corresponding conclusions in the literature to ${\mathbb{R}}^N$ for any $N\ge 3$.

2 A lower bound for the blow-up time

In this section we seek the lower bound for the blow-up time $t^{\ast }$ and establish the following theorem.

Theorem 1

Let$u(x,t)$be the classical nonnegative solution of problem (1.1)-(1.3) in a bounded star-shaped domain$\mathrm{\Omega }\in {\mathbb{R}}^N$ ($N\ge 3$) and assume that$q<p$. Then the quantity

$$\phi (t)={\int }_{\mathrm{\Omega }}{u}^{ns}dx$$

(2.1)

satisfies the differential inequality

$$\frac{d\phi }{dt}\le k_1{\phi }^{1+\beta },$$

(2.2)

from which follows that the blow-up time$t^{\ast }$is bounded from below; i.e., we have

$$t^{\ast }\ge {\int }_{\phi (0)}^{\mathrm{\infty }}\frac{1}{k_1{\eta }^{1+\beta }}d\eta =\frac{1}{k_1\beta }{\phi }^{-\beta }(0),$$

(2.3)

where$k_1$, β are positive constants which will be defined later.

Proof

Firstly we compute

$$\begin{array}{rl}{\phi }^{\prime }(t)& =ns{\int }_{\mathrm{\Omega }}{u}^{ns-1}[\mathrm{△}u+{\int }_{\mathrm{\Omega }}{u}^{s+1}dx-k{u}^q]dx\\ \le -\frac{4(ns-1)}{ns}{\int }_{\mathrm{\Omega }}{\left|\mathrm{\nabla }{u}^{\frac{ns}{2}}\right|}^{2}dx+ns|\mathrm{\Omega }|{\int }_{\mathrm{\Omega }}{u}^{s(n+1)}dx-kns{\int }_{\mathrm{\Omega }}{u}^{ns+q-1}dx.\end{array}$$

(2.4)

For convenience, we now set

$$v=u^s,\alpha =\frac{q-1}{s}.$$

(2.5)

Since $q<s+1$, $\alpha <1$. Thus, we obtain

$${\phi }^{\prime }(t)\le -\frac{4(ns-1)}{ns}{\int }_{\mathrm{\Omega }}{\left|\mathrm{\nabla }{v}^{\frac{n}{2}}\right|}^{2}dx+ns|\mathrm{\Omega }|{\int }_{\mathrm{\Omega }}{v}^{(n+1)}dx-kns{\int }_{\mathrm{\Omega }}{v}^{n+\alpha }dx.$$

(2.6)

By the Hölder inequality, we have

$${\int }_{\mathrm{\Omega }}{v}^{n+1}dx\le {\left({\int }_{\mathrm{\Omega }}{v}^{n+\alpha }dx\right)}^{\frac{\gamma -1}{\gamma -\alpha }}{\left({\int }_{\mathrm{\Omega }}{v}^{n+\gamma }dx\right)}^{\frac{1-\alpha }{\gamma -\alpha }}$$

(2.7)

for positive constant $\gamma >1$. By the inequality

$$a^r+b^{1-r}\le ra+(1-r)b,a,b>0,0<r<1,$$

(2.8)

we have

$${\int }_{\mathrm{\Omega }}{v}^{n+1}dx\le \frac{\gamma -1}{\gamma -\alpha }{\epsilon }_1{\int }_{\mathrm{\Omega }}{v}^{n+\alpha }dx+\frac{1-\alpha }{\gamma -\alpha }{\epsilon }_1^{-\frac{\gamma -1}{1-\alpha }}{\int }_{\mathrm{\Omega }}{v}^{n+\gamma }dx,$$

(2.9)

where ${\epsilon }_1$ is a positive constant. If we insert (2.9) into (2.6) and choose

$${\epsilon }_1=\frac{k(\gamma -\alpha )}{|\mathrm{\Omega }|(\gamma -1)},$$

then (2.6) yields

$${\phi }^{\prime }(t)\le -\frac{4(ns-1)}{ns}{\int }_{\mathrm{\Omega }}{\left|\mathrm{\nabla }{v}^{\frac{n}{2}}\right|}^{2}dx+ns|\mathrm{\Omega }|\frac{1-\alpha }{\gamma -\alpha }{\epsilon }_1^{-\frac{\gamma -1}{1-\alpha }}{\int }_{\mathrm{\Omega }}{v}^{n+\gamma }dx.$$

(2.10)

By the Hölder inequality again, we have

$${\int }_{\mathrm{\Omega }}{v}^{n+\gamma }dx\le {\left({\int }_{\mathrm{\Omega }}{v}^{n}dx\right)}^{\frac{2n-\gamma (N-2)}{2n}}{\left({\int }_{\mathrm{\Omega }}{v}^{\frac{n}{2}\cdot \frac{2N}{N-2}}dx\right)}^{\frac{\gamma (N-2)}{2n}},$$

(2.11)

where we have chosen $2n>\gamma N$. Now let $c_1$ be the best imbedding constant defined in [27]. Using the Sobolev inequality for $W_0^{1,2}\hookrightarrow L^{\frac{2N}{N-2}}$ for $N\ge 3$, we have

$${\int }_{\mathrm{\Omega }}{v}^{\frac{n}{2}\cdot \frac{2N}{N-2}}dx\le c_1^{\frac{2N}{N-2}}{\left({\int }_{\mathrm{\Omega }}{\left|\mathrm{\nabla }{v}^{\frac{n}{2}}\right|}^{2}dx\right)}^{\frac{N}{N-2}}.$$

Therefore, (2.11) may be rewritten as

$${\int }_{\mathrm{\Omega }}{v}^{n+\gamma }dx\le c_1^{\frac{\gamma N}{2n}}{\left({\int }_{\mathrm{\Omega }}{v}^{n}dx\right)}^{\frac{2n-\gamma (N-2)}{2n}}{\left({\int }_{\mathrm{\Omega }}{\left|\mathrm{\nabla }{v}^{\frac{n}{2}}\right|}^{2}dx\right)}^{\frac{N\gamma }{2n}}.$$

(2.12)

Using (2.8) again, we have

$$\begin{array}{rl}{\int }_{\mathrm{\Omega }}{v}^{n+\gamma }dx\le & \frac{2n-N\gamma }{2n}{c}_1^{\frac{\gamma N}{2n}}{\epsilon }_2^{-\frac{N}{2n-\gamma (N-2)}}{\left({\int }_{\mathrm{\Omega }}{v}^{n}dx\right)}^{\frac{2n-\gamma (N-2)}{2n}}\\ +\frac{N\gamma }{2n}{c}_1^{\frac{\gamma N}{2n-N\gamma }}{\epsilon }_2{\int }_{\mathrm{\Omega }}{\left|\mathrm{\nabla }{v}^{\frac{n}{2}}\right|}^{2}dx,\end{array}$$

(2.13)

where ${\epsilon }_2$ is a positive constant to be chosen as follows:

$${\epsilon }_2=\frac{8(ns-1)(\gamma -\alpha )}{N\gamma |\mathrm{\Omega }|n{s}^2(1-\alpha )}{\epsilon }_1^{\frac{\gamma -1}{1-\alpha }}{c}_1^{-\frac{\gamma N}{2n}},$$

(2.14)

and inserting (2.13) back into (2.10), we have

$${\phi }^{\prime }(t)\le k_1{\phi }^{\frac{2n-\gamma (N-2)}{2n-N\gamma }},$$

(2.15)

where

$$k_1=ns|\mathrm{\Omega }|\frac{1-\alpha }{\gamma -\alpha }{\epsilon }_1^{-\frac{\gamma -1}{1-\alpha }}\frac{2n-N\gamma }{2n}{c}_1^{\frac{\gamma N}{2n}}{\epsilon }_2^{-\frac{N}{2n-\gamma (N-2)}}.$$

(2.16)

If we set

$$\beta =\frac{2\gamma }{2n-N\gamma }>0,$$

(2.17)

then (2.15) can be written as

$${\phi }^{\prime }(t)\le k_1{\phi }^{1+\beta },$$

(2.18)

or

$$\frac{d\phi }{k_1{\phi }^{1+\beta }}\le 1.$$

(2.19)

Upon integration we have for $t<t^{\ast }$,

$$t^{\ast }\ge {\int }_{\phi (0)}^{\mathrm{\infty }}\frac{1}{k_1{\eta }^{1+\beta }}d\eta =\frac{1}{k_1\beta }{\phi }^{-\beta }(0),$$

(2.20)

where $\phi (0)=\phi (t)={\int }_{\mathrm{\Omega }}{u}_0^{ns}dx$. □