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:reaction-diffusion; absorption; blow-up; blow-up rate; non-extinction
In this paper, we consider the reaction-diffusion equations with absorption
where , , , , is a bounded domain with smooth boundary ∂Ω, and 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 () of (1.1) has been investigated by Bedjaoui and Souplet . They obtained that the solutions exist globally if either or , and the solutions may blow up in finite time for large initial value if . Recently, Xiang et al. considered the blow-up rate estimates for nonglobal solutions of (1.1) () with , and obtained that (i) ; (ii) if , where are positive constants. Liu et al. studied the extinction phenomenon of solutions of (1.1) for the case with and obtained some sufficient conditions about the extinction in finite time and decay estimates of solutions in ().
Recently, Zhou et al. investigated positive solutions of the degenerate parabolic equation not in divergence form
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).
In particular, is called a weak solution of (1.1) if it is both a weak upper and a weak lower solution. For every , if is a weak solution of (1.1) in , we say that is global. The local in time existence of nonnegative weak solutions have been established (see the survey ), 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 -parameter region into three classes: (i) ; (ii) ; (iii) .
We remark that in , Liang  studied the blow up rate of blow-up solutions to the following Cauchy problem
Now, we pay attention to the nonextinction property of solutions and have the following result.
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
By using Hölder’s inequality, we discover
Inserting (2.4) into (2.3), we have
According to (2.5), (2.6), we obtain
as long as
Now deal with the nonexistence of global solutions, we seek a blow-up self-similar lower solution of the problem (1.1). Without loss of generality, we may assume that Ω contains the origin. Since , we can choose constant α such that
and consider the function
After some computations, we have
It will be obtained from the above equalities that
It is easy to see that
For , we have . It follow from that (2.9) is satisfied for , if T is sufficiently small. Therefore, given by (2.8) is a blow-up lower solution of the problem (1.1) with appropriately large . 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 and , , here . Then we know from the assumption and comparison principle that u is radially decreasing in r with . In this section, denote by T the blow-up time for the nonglobal solutions to (1.1).
We now construct an upper solution for this problem. Set
and the lower estimate is obtained.
Now, in order to deal with the initial data, consider the function
After a direct computation, we have
and the upper estimate is obtained. □
We discuss the nonextinction of the solution to the problem (1.1) in this section. For , 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 , there exists a region and such that a.e. in . is the first Dirichlet eigenvalue of −Δ on with corresponding eigenfunction , normalized by , and prolong solution by 0 in . We treat the five subcases for the proof.
Theorem A.1 (Comparison principle)
By a simple calculation, we have
where L is a positive constant. By (A.1), (A.2), we have
It follows immediately by using the Gronwall’s inequality that
The authors declare that they have no competing interests.
DW carried out all studies in the paper. LZ participated in the design of the study in the paper.
This work was partially supported by Projects Supported by Scientific Research Fund of Sichuan Provincial Education Department (09ZA119).
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