# Asymptotic analysis for reaction-diffusion equations with absorption

- Wanjuan Du
^{1}Email author and - Zhongping Li
^{1}

**2012**:84

https://doi.org/10.1186/1687-2770-2012-84

© Du and Li; licensee Springer 2012

**Received: **31 May 2012

**Accepted: **20 July 2012

**Published: **2 August 2012

## Abstract

In this paper, we study the blow-up and nonextinction phenomenon of reaction-diffusion equations with absorption under the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then give the blow-up rate estimates for the nonglobal solutions. In addition, the nonextinction of solutions is also concerned.

**MSC:**35B33, 35K55, 35K60.

## Keywords

## 1 Introduction

where $m>1$, $p>0$, $q\ge 1$, $p\ne q$, $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$ is a bounded domain with smooth boundary *∂* Ω, and ${u}_{0}(x)$ is a nontrivial, nonnegative, bounded, and appropriately smooth function. Parabolic equations like (1.1) appear in population dynamics, chemical reactions, heat transfer, and so on. We refer to [2, 8, 9] for details on physical models involving more general reaction-diffusion equations.

The semilinear case ($m=1$) of (1.1) has been investigated by Bedjaoui and Souplet [3]. They obtained that the solutions exist globally if either $p<max\{q,1\}$ or $p=max\{q,1\}$, and the solutions may blow up in finite time for large initial value if $p>max\{q,1\}$. Recently, Xiang *et al.* [11] considered the blow-up rate estimates for nonglobal solutions of (1.1) ($m=1$) with $p>max\{q,1\}$, and obtained that (i) ${max}_{\mathrm{\Omega}}u(x,t)\ge c{(T-t)}^{-\frac{1}{p-1}}$; (ii) ${max}_{\mathrm{\Omega}}u(x,t)\le C{(T-t)}^{-\frac{1}{p-1}}$ if $p\le 1+\frac{2}{N+1}$, where $c,C>0$ are positive constants. Liu *et al.* [7] studied the extinction phenomenon of solutions of (1.1) for the case $0<m<1$ with $q=1$ and obtained some sufficient conditions about the extinction in finite time and decay estimates of solutions in $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$ ($N>2$).

*et al.*[10] investigated positive solutions of the degenerate parabolic equation not in divergence form

where $p\ge 1$, $q,a,b>0$, $r>1$. They at first gave some conditions about the existence and nonexistence of global solutions to (1.2), and then studied the large time behavior for the global solutions.

Motivated by the above mentioned works, the aim of this paper is threefold. First, we determine optimal conditions for the existence and nonexistence of global solutions to (1.1). Secondly, by using the scaling arguments we establish the exact blow-up rate estimates for solutions which blow up in a finite time. Finally, we prove that every solution to (1.1) is nonextinction.

As it is well known that degenerate equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).

**Definition 1.1**Let $T>0$ and ${Q}_{T}=\mathrm{\Omega}\times (0,T)$, $E=\{u\in {L}^{2p}({Q}_{T})\cap {L}^{2q}({Q}_{T});{u}_{t},\mathrm{\nabla}u\in {L}^{2}({Q}_{T})\}$, ${E}_{0}=\{u\in E;u=0\text{on}\partial \mathrm{\Omega}\}$, a nonnegative function $u(x,t)\in E$ is called a weak upper (or lower) solution to (1.1) in ${Q}_{T}$ if for any nonnegative function $\phi \in {E}_{0}$, one has

In particular, $u(x,t)$ is called a weak solution of (1.1) if it is both a weak upper and a weak lower solution. For every $T<\mathrm{\infty}$, if $u(x,t)$ is a weak solution of (1.1) in ${Q}_{T}$, we say that $u(x,t)$ is global. The local in time existence of nonnegative weak solutions have been established (see the survey [1]), and the weak comparison principle is stated and proved in the Appendix in this paper.

The behavior of the weak solutions is determined by the interactions among the multinonlinear mechanisms in the nonlinear diffusion equations in (1.1). We divide the $(m,p,q)$-parameter region into three classes: (i) $p<max\{m,q\}$; (ii) $p=max\{m,q\}$; (iii) $p>max\{m,q\}$.

**Theorem 1.1** *If* $p<max\{m,q\}$, *then all solutions of* (1.1) *are bounded*.

with the first eigenvalue ${\lambda}_{1}$, normalized by ${\parallel \varphi \parallel}_{\mathrm{\infty}}=1$, then ${\lambda}_{1}>0$ and $\varphi >0$ in Ω.

**Theorem 1.2** *Assume that* $p=max\{m,q\}$. *Then all solutions are global if* ${\lambda}_{1}\ge 1$, *and there exist both global and nonglobal solutions if* ${\lambda}_{1}<1$.

**Theorem 1.3** *If* $p>max\{m,q\}$, *then there exist both global and nonglobal solutions to* (1.1).

To obtain the blow-up rate of blow-up solutions to (1.1), we need an extra assumption that $\mathrm{\Omega}={B}_{R}(0)=\{x\in {\mathbb{R}}^{N}:|x|<R\}$ and ${u}_{0}={u}_{0}(r)$, ${u}_{0}^{\prime}(r)\le 0$, here $r=|x|$. By the assumption and comparison principle, we know that *u* is radially decreasing in *r* with ${max}_{\mathrm{\Omega}}u(x,t)=u(0,t)$.

**Theorem 1.4**

*Suppose that*$p>max\{m,q\}$.

*If the solution*$u(x,t)$

*of*(1.1)

*blows up in finite time*

*T*,

*then there exists a positive constant*

*c*

*such that*

*Furthermore*,

*if*$p>m\ge q$,

*then we have also the upper estimate*,

*that is*,

*there exists a positive constant*

*C*

*such that*

with the bounded initial function, $1<m<p<m\frac{N+2}{{(N-2)}_{+}}$, and obtained that ${\parallel u\parallel}_{{L}^{\mathrm{\infty}}({\mathbb{R}}^{N})}<C{(T-t)}^{-\frac{1}{p-1}}$ for $t\in (0,T)$. By using the same scaling arguments in this paper, we can find that Theorem 1.4 is correct for (1.4) with $p>m$.

Now, we pay attention to the nonextinction property of solutions and have the following result.

**Theorem 1.5** *Any solution of* (1.1) *does not go extinct in finite time for any nontrivial and nonnegative initial value* ${u}_{0}(x)$ *with* $meas\{x\in \mathrm{\Omega};{u}_{0}(x)>0\}>0$.

The rest of this paper is organized as follows. In the next section, we discuss the global existence and nonexistence of solutions, and prove Theorems 1.1-1.3. Subsequently, in Sects. 3 and 4, we consider the estimate of the blow-up rate and study the nonextinction phenomenon for the problem (1.1). The weak comparison principle is stated and proved in the Appendix.

## 2 Global existence and nonexistence

*Proof of Theorem 1.1*If $m\ge q$, that is $p<m$, then by the comparison principle, we have $u\le w$, where

*w*satisfies

We know from [4, 5] that *w* is bounded.

If $m<q$, we have $p<q$. It is obvious that $\overline{u}=max\{1,{\parallel {u}_{0}\parallel}_{\mathrm{\infty}}\}$ is a time-independent upper solution to (1.1). □

*Proof of Theorem 1.2* Since $p\ne q$ and $p=max\{m,q\}$ imply $p=m>q$. Due to the fact that the solution of (2.1) is an upper solution of (1.1), the conclusions for ${\lambda}_{1}\ge 1$ is obvious true; see [4, 5].

provided ${h}_{0}^{m-q}{\psi}^{\frac{m-q}{m}}\le 1$. Thus, $\overline{u}$ is an upper solution of (1.1), and consequently, $u\le \overline{u}=h(t){\psi}^{\frac{1}{m}}(x)\to 0$ as $t\to \mathrm{\infty}$.

where $r=\frac{m-1}{m}$, $s=\frac{q}{m}<1$ and ${v}_{0}(x)={u}_{0}^{m}(x)$.

*ϕ*is given in (1.3). Then we have

*i.e.*,

we then follow from (2.7) that $J(\tau )$, and consequently $u(x,t)$, blows up in finite time since $J(\tau )$ is increasing and $\frac{1}{1-r}=m>1$. □

*Proof of Theorem 1.3*Let $h(t)$ solves ${h}^{\prime}(t)=-h{(t)}^{p}$ with $h(0)={h}_{0}$, and set $\overline{u}=h(t){\psi}^{\frac{1}{m}}(x)$, where

*ψ*is defined in (2.2). Then

Since $p>max\{m,q\}$, we can choose ${h}_{0}$ small enough such that ${\overline{u}}_{t}-\mathrm{\Delta}{\overline{u}}^{m}-{\overline{u}}^{p}+{\overline{u}}^{q}\ge 0$. Thus, $\overline{u}$ is an upper solution of (1.1) provided ${u}_{0}(x)\le {h}_{0}{\psi}^{\frac{1}{m}}(x)$, and consequently, $u\le \overline{u}=h(t){\psi}^{\frac{1}{m}}(x)\to 0$ as $t\to \mathrm{\infty}$.

*α*such that

where $f(\xi )={({a}^{2}-{\xi}^{2})}_{+}^{\frac{1}{m-1}}$. Note that the support of $\underline{u}(x,t)$ is contained in $B(0,a{T}^{\beta})$, which is included in Ω if *T* is sufficiently small.

For $0<\xi \le \theta a$, we have $f(\xi )\ge {(1-{\theta}^{2})}^{\frac{1}{m-1}}{a}^{\frac{2}{m-1}}>0$. It follow from $p\alpha >\alpha +1>q\alpha $ that (2.9) is satisfied for $0<\xi \le \theta a$, $\theta a<\xi <a$ if *T* is sufficiently small. Therefore, $\underline{u}$ given by (2.8) is a blow-up lower solution of the problem (1.1) with appropriately large ${u}_{0}$. And consequently, there exist nonglobal solutions to (1.1). □

## 3 Blow-up rate

In this section, we study the speeds at which the solutions to (1.1) blow up. Assume that $\mathrm{\Omega}={B}_{R}(0)=\{x\in {\mathbb{R}}^{N}:|x|<R\}$ and ${u}_{0}={u}_{0}(r)$, ${u}_{0}^{\prime}(r)\le 0$, here $r=|x|$. Then we know from the assumption and comparison principle that *u* is radially decreasing in *r* with ${max}_{\mathrm{\Omega}}u(x,t)=u(0,t)$. In this section, denote by *T* the blow-up time for the nonglobal solutions to (1.1).

*Proof of Theorem 1.4*Fix $t\in (0,T)$ such that $M(t)={max}_{\mathrm{\Omega}}u(x,t)\ge 1$, and let

*M*, for all

*M*large enough. Therefore, the blow-up time of ${\psi}_{M}$ is greater than ${S}_{1}$, that is ${M}^{p-1}(T-t)\ge {S}_{1}$. This implies

and the lower estimate is obtained.

*θ*is given in (2.10). Let ${M}_{0}$ satisfies

*μ*are to be determined later. Clearly, $\underline{w}(y,s)=0$ on $\partial {B}_{{M}^{\frac{p-m}{2}}R}(0)\times (0,S)$. As the same arguments in the proof of Theorem 1.3, we have for $\theta a\le \xi <a$ that

where $\lambda =\frac{1}{(m-1)N+1}$, $\mu =\frac{1-\lambda}{m-1}$, and ${S}_{2}=\frac{4m}{(m-1)\lambda}$.

Furthermore, $z(y,s)=0$ on $\partial {B}_{{M}^{\frac{p-m}{2}}R}(0)\times (0,1)$. In addition, $z(y,0)={lim}_{s\to 0}z(y,s)=0$ a.e. in ${B}_{{M}^{\frac{p-m}{2}}R}(0)$. Therefore, by the comparison principle, we have that ${\psi}_{M}(y,s)\ge z(y,s)$ for $0\le s\le 1$. By the virtue of $\underline{w}(y,0)=z(y,1)$, we have ${\psi}_{M}(y,s+1)\ge \underline{w}(y,s)$.

*M*, for all $M>{M}_{0}$. Therefore, the blow-up time of ${\psi}_{M}$ is less than ${S}_{2}+1$, that is ${M}^{p-1}(T-t)\le {S}_{2}+1$. This implies

and the upper estimate is obtained. □

## 4 Nonextinction

We discuss the nonextinction of the solution to the problem (1.1) in this section. For $p<1$, the uniqueness of the weak solution to (1.1) may not hold. In this case, we only consider the maximal solution, which can be obtained by standard regularized approximation methods. Clearly, the comparison principle is valid for the maximal solution.

*Proof of Theorem 1.5*For $meas\{x\in \mathrm{\Omega};{u}_{0}(x)>0\}>0$, there exists a region ${\mathrm{\Omega}}_{0}\subset \mathrm{\Omega}$ and $\u03f5\in (0,1)$ such that ${u}_{0}(x)\ge \u03f5$ a.e. in ${\mathrm{\Omega}}_{0}$. ${\lambda}_{0}$ is the first Dirichlet eigenvalue of −Δ on ${\mathrm{\Omega}}_{0}$ with corresponding eigenfunction ${\varphi}_{0}(x)$, normalized by ${\parallel {\varphi}_{0}\parallel}_{\mathrm{\infty}}=1$, and prolong solution ${\varphi}_{0}$ by 0 in $\mathrm{\Omega}\setminus {\mathrm{\Omega}}_{0}$. We treat the five subcases for the proof.

- (a)

- (b)For $p>1$, $1\le q\le m$, we let $\underline{u}=h(t){\varphi}_{0}^{\frac{1}{m}}$, ${h}^{\prime}(t)=-(1+{\lambda}_{0}){h}^{q}(t)$ with $h(0)={h}_{0}<\u03f5$. Then$\begin{array}{rcl}{\underline{u}}_{t}-\mathrm{\Delta}{\underline{u}}^{m}-{\underline{u}}^{p}+{\underline{u}}^{q}& \le & -(1+{\lambda}_{0}){h}^{q}(t){\varphi}_{0}^{\frac{1}{m}}+{\lambda}_{0}{h}^{m}{\varphi}_{0}+{h}^{q}{\varphi}_{0}^{\frac{q}{m}}\\ =& -{h}^{q}(t){\varphi}_{0}^{\frac{1}{m}}(1+{\lambda}_{0}-{\lambda}_{0}{h}^{m-q}{\varphi}_{0}^{\frac{m-q}{m}}-{\varphi}_{0}^{\frac{q-1}{m}})\\ \le & 0.\end{array}$

- (c)

- (d)

- (e)

*c*satisfies $1+{\lambda}_{0}<c<1+{\lambda}_{0}+{\u03f5}^{p-m}$. It is easy to see that $h(t)$ is nonincreasing and $h(t)\to 0$ as $t\to \mathrm{\infty}$.

By the comparison principle, we have $u\ge \underline{u}>0$ in ${\mathrm{\Omega}}_{0}$.

□

## Appendix

**Theorem A.1** (Comparison principle)

*Let* $\underline{u}$ *and* $\overline{u}$ *are a weak lower and a weak upper solutions of* (1.1) *in* ${Q}_{T}$. *If* $p\ge 1$ *or* $\overline{u}$ *has a positive lower bound*, *then* $\underline{u}\le \overline{u}$ *a*.*e*. *in* ${Q}_{T}$.

*Proof*From the definition of weak upper and lower solutions, for any $0\le \phi \in {E}_{0}$, we obtain

*L*is a positive constant. By (A.1), (A.2), we have

for almost all $t\in (0,T)$, and hence $\underline{u}\le \overline{u}$ a.e. in $\mathrm{\Omega}\times (0,T)$. □

## Declarations

### Acknowledgements

This work was partially supported by Projects Supported by Scientific Research Fund of Sichuan Provincial Education Department (09ZA119).

## Authors’ Affiliations

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