Asymptotic analysis for reaction-diffusion equations with absorption
© Du and Li; licensee Springer 2012
Received: 31 May 2012
Accepted: 20 July 2012
Published: 2 August 2012
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.
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 ().
where , , . 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).
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) .
Theorem 1.1 If , then all solutions of (1.1) are bounded.
with the first eigenvalue , normalized by , then and in Ω.
Theorem 1.2 Assume that . Then all solutions are global if , and there exist both global and nonglobal solutions if .
Theorem 1.3 If , 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 and , , here . By the assumption and comparison principle, we know that u is radially decreasing in r with .
with the bounded initial function, , and obtained that for . By using the same scaling arguments in this paper, we can find that Theorem 1.4 is correct for (1.4) with .
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 with .
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
If , we have . It is obvious that is a time-independent upper solution to (1.1). □
provided . Thus, is an upper solution of (1.1), and consequently, as .
where , and .
we then follow from (2.7) that , and consequently , blows up in finite time since is increasing and . □
Since , we can choose small enough such that . Thus, is an upper solution of (1.1) provided , and consequently, as .
where . Note that the support of is contained in , which is included in Ω if T is sufficiently small.
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).
and the lower estimate is obtained.
where , , and .
Furthermore, on . In addition, a.e. in . Therefore, by the comparison principle, we have that for . By the virtue of , 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.
- (b)For , , we let , with . Then
By the comparison principle, we have in .
Theorem A.1 (Comparison principle)
Let and are a weak lower and a weak upper solutions of (1.1) in . If or has a positive lower bound, then a.e. in .
for almost all , and hence a.e. in . □
This work was partially supported by Projects Supported by Scientific Research Fund of Sichuan Provincial Education Department (09ZA119).
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