The boundary quenching behavior of a semilinear parabolic equation

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Abstract

In this paper we consider the boundary quenching behavior of a semilinear parabolic problem in one-dimensional space, of which the nonlinearity appears both in the source term and in the Neumann boundary condition. First we proved that the solution quenches at somewhere in some finite time. Then we assert that the quenching can only occur on the left boundary if the given initial datum is monotone. Finally we derived the upper and lower bounds for the quenching rate of the solution near the quenching time. Thus we generalized our former results.

Introduction

In this paper the quenching behavior of a one-dimensional semilinear parabolic equation with nonlinear boundary outfluxut=uxx+f(x)(1-u)-p,0<x<1,0<t<,ux(0,t)=u-q(0,t),ux(1,t)=0,0<t<,u(x,0)=u0(x),0x1is considered, where p,q are positive constants, the initial datum u0:[0,1](0,1) is smooth enough and satisfies compatibility condition on the lateral boundary, and f(x) is non-negative. This problem can be considered as a heat conduction model that incorporates the effects of nonlinear reaction (source) and nonlinear boundary outflux (emission).

As to the term of quenching, there have been mainly two kinds of definitions: The former requires that the solution approaches a constant but its derivative with respect to the time variable t tends to infinity, as (x,t) tends to some point in spatial-time space (for details see [2], [3]). The latter, which is now prevailing, requires only the solution tends to a constant, and in some cases is equivalent to the former (for details reference [4], [5]). In this paper we take the latter and thus give the following definition.

Definition 1

We say that the solution u(x,t) to the problem (1) quenches in finite time T, if there exists 0<T<,such thatlimtT-min0x1u(x,t)=0orlimtT-max0x1u(x,t)=1.

From now on, we denote by T(0<T<) the quenching time of the problem (1).

Some authors have discussed such quenching problems with which the nonlinearities appear both in source (or sink) and in boundary conditions (see [6] and references therein). Zhao [6] considered a parabolic quenching problem as followsut=Δu+up,xΩ,t>0,uν=-u-q,xΩ,t>0,u(x,0)=u0(x),xΩand showed that the quenching can only occur on the boundary under some conditions upon the initial datum. He also gave the quenching rate estimates which is (T-t)1/(2(q+1)) if T denotes the quenching time. Zhao’s work motivates our former paper [1], and in this paper we want to generalize the results of [1] to the case of more general reaction (source) term.

Observe that in our problem (1) the nonlinear source term may become singular if u(x,t)1as(x,t)(xˆ,tˆ),wherexˆ,tˆ is a point in [0,1]×(0,). On the other hand, the outflux u-q(0,t) may also become singular in some finite time. If these two cases may happen, it is certainly hard to deal with, for we must judge which term becomes singular first, f(x)(1-u)-p or u-q(0,t), or simultaneously. At present few author has considered this problem. We will preclude the formation of the singularity for the source term, provided the initial datum satisfies some conditions, thus we try to consider the boundary quenching behavior of the solution to problem (1): to determine the quenching set and to derive the quenching rate estimates. Thus we can depict if and how the function f(x) affects the boundary quenching behavior.

Now we give a list of our main results of this paper.

Theorem 1

Assume that the initial datum satisfiesu0+f(x)(1-u0(x))-p0and not equals 0 identicallyfor 0<x<1. Then there exists a finite time T, such that the solution u to the problem (1) quenches in this time.

The following theorem is about quenching set, that is the set{xˆ[0,1]|limtT-u(xˆ,t)=0orlimtT-u(xˆ,t)=1}.

Theorem 2

Assume (3) remains true. Then the quenching can only take place on the boundary of interval (0,1).

Remark 1

By noticing condition (3) and using the Maximum Principle we see ut(x,t)0 in [0,1]×(0,T), which leads immediately to the impossibility of the formation of the singularity in time T for the source term.

By Theorem 2 and the Maximum Principle we have.

Corollary 1

Assume that the initial datum satisfiesu0(x)0,u0(x)+f(x)(1-u0(x))-p0,in (0,1). Then the quenching can only occur at the point x=0 in finite time T.

Remark 2

From Corollary 1, we see that in Definition 1 the case oflimtT-max0x1u(x,t)=1will not occur because of our choice of the initial datum.

Remark 3

Actually the conditions of (4) is proper because we can easily construct such a function (initial datum) satisfying (4). For example, if we let p=1,f(x)=1, then we can easily check thatu0(x)2-121/2x+2-121/2,satisfies the conditions of (4).

The two theorems below give the lower and upper bound for the solution u near the quenching time T.

At first, by Walter’s method of inequalities (see [2]), we proved the existence of a lower bound for the quenching rate.

Theorem 3

There exists a positive constant C1 such thatu(0,t)C1(T-t)12(q+1)astclose from below toT,provided u0(x)0,f(x)0 and u0(x)+f(x)(1-u0(x))-p0in(0,1).

In order to obtain the upper bound for the quenching rate of the solution of the problem under considering, it is in some cases useful to transform the quenching problem to a corresponding blow-up problem by set u=1w (for details see [6]). But to our problem (1), just as in [1], we can not use this method of transformation directly, but prefer to work with problem (1) directly, thus derived following Theorem concerning the upper bound for the quenching rate.

Theorem 4

Assume that u0(x)0,f(x)0,u0(x)+(x)(1-u0(x))-p0. Then there exists a constant C2>0, such thatu(0,t)C2(T-t)12(q+1)astclose from below toT.

Combining Theorem 3, Theorem 4 we have.

Corollary 2

Assume that u0(x)0,f(x)0,u0(x)+(x)(1-u0(x))-p0. Then, near the quenching time T, the solution u(x,t) to problem Eq. (1) has following quenching rate estimateu(0,t)(T-t)12(q+1).

Remark 4

From Theorem 4 we see that the upper bound for the quenching rate remains the same as that of [5], but the initial datum there was assumed to be smoother than here. By comparing the problem (1) with (2), we can see from Corollary 2 that although the nonlinearities appears also in the source f(x)(1-u)-p the quenching rates still remain the same as that of [7] and of [1], [6]. Therefore the nonlinearity of source term, especially the occurrence of f(x), has in fact no essential effect upon the quenching behavior of problem (1), thus the results here generalized to some extent the results of [1].

The rest of this paper is organized as follows. In the next section we prove Theorem 1, Theorem 2; The proofs of Theorem 3, Theorem 4 are given in Section 3.

Section snippets

On the quenching set

In this section, we prove that the quenching occurs in some finite time T, provided the initial datum satisfies (3), and that the quenching set is the singleton {x=0}.

We first give the proof of Theorem 1.

Proof of Theorem 1

We prove this result by contradiction. Assume on the contrary u can not quenches at all time. Setγu0-q(0)-01f(x)(1-u0(x))-pdx,then clearly γ>0 according to (3).

Now introduce a mass function: F(t)=01u(x,t)dx,0<t<T. ThenF(t)-(u0(0))-q+01f(x)(1-u0(x))-pdx=-γ,considering the condition (3) and

On the estimates of quenching rate

In this section we want to give the quenching rate estimates for the solution of our problem. But in order to give the Proof of Theorem 3, Proof of Theorem 4, we need a lower bound for ux.

Lemma 2

If u0(x)0 and f(x)0, then there exists a positive constant C6, such thatux(x,t)C6inQ1[0,1/2]×[T/2,T).

Proof

Let h(x,t)ux+μx-34, where μ>0 will be specified later. Then we have h(x,T/2)0 if μ is small, and h(3/4,t)0 as T/2t<T,h(0,t)=u-q(0,t)-μ340 if μ is small, and as wellht-hxx-p(1-u)-p-1f(x)h=f(x)(1-u)-p-

Acknowledgements

This work is supported by YNU Postdoctoral Science Foundation Project (W4030002) and Science Foundation of Yunnan University (2007Q019C).

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