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
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).
Definition 1.1 Let and , , , a nonnegative function is called a weak upper (or lower) solution to (1.1) in if for any nonnegative function , one has
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 1.1If , then all solutions of (1.1) are bounded.
Let be the first eigenfunction of
with the first eigenvalue , normalized by , then and in Ω.
Theorem 1.2Assume that . Then all solutions are global if , and there exist both global and nonglobal solutions if .
Theorem 1.3If , 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 .
Theorem 1.4Suppose that . If the solution of (1.1) blows up in finite timeT, then there exists a positive constantcsuch that
Furthermore, if , then we have also the upper estimate, that is, there exists a positive constantCsuch that
We remark that in , Liang  studied the blow up rate of blow-up solutions to the following Cauchy problem
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 1.5Any 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
Proof of Theorem 1.1 If , that is , then by the comparison principle, we have , where w satisfies
If , we have . It is obvious that is a time-independent upper solution to (1.1). □
Now consider with small initial data. Let be the unique solution of
and solves with , where . Set . Then
provided . Thus, is an upper solution of (1.1), and consequently, as .
If with large initial data, we first introduce some transformations. Let and , then (1.1) becomes the following equations not in divergence form:
where , and .
Let , where ϕ is given in (1.3). Then we have
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
Hence, if satisfies
we then follow from (2.7) that , and consequently , blows up in finite time since is increasing and . □
Proof of Theorem 1.3 Let solves with , and set , where ψ is defined in (2.2). Then
Since , we can choose small enough such that . Thus, is an upper solution of (1.1) provided , and consequently, 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
where . Note that the support of is contained in , which is included in Ω if T is sufficiently small.
After some computations, we have
It will be obtained from the above equalities that
It is easy to see that
To satisfy (2.9), we distinguish the two zones and , where
For , we have
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).
Proof of Theorem 1.4 Fix such that , and let
and define the function in , where . blows up at , moreover, it is a solution of the following problem:
We now construct an upper solution for this problem. Set
where , , and
After a direct computation, for , we have
Clearly, for , and on , in . We have an upper solution independent of M, for all M large enough. Therefore, the blow-up time of is greater than , that is . This implies
and the lower estimate is obtained.
In order to obtained the upper estimates for the blow-up rate, we look for a lower solution to (3.1) with . Set
where , , , and θ is given in (2.10). Let satisfies
where , μ are to be determined later. Clearly, on . As the same arguments in the proof of Theorem 1.3, we have for that
For , we have that
Now, in order to deal with the initial data, consider the function
where , , and .
After a direct computation, we have
Furthermore, on . In addition, a.e. in . Therefore, by the comparison principle, we have that for . By the virtue of , we have .
We have a lower solution independent of M, for all . Therefore, the blow-up time of is less than , that is . This implies
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.
(a) For , set , where
By the comparison principle, we have in .
(b) For , , we let , with . Then
Then we know by the comparison principle that in .
(c) For , , we let , and
where . It is easy to see that is nonincreasing and as .
And consequently, in .
(d) For , , we let , where
Obviously, is nonincreasing and as .
Thus, we have in .
(e) For , , we let , and
where c satisfies . It is easy to see that is nonincreasing and as .
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 .
Proof From the definition of weak upper and lower solutions, for any , we obtain
Let for . Choose , where is the characteristic function defined on , Then we arrive at
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
for almost all , and hence a.e. in . □
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).
Aronson, D, Crandall, MG, Peletier, LA: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal.. 6, 1001–1022 (1982). Publisher Full Text
Bedjaoui, N, Souplet, Ph: Critical blowup exponents for a system of reaction-diffusion equations with absorption. Z. Angew. Math. Phys.. 53, 197–210 (2002). Publisher Full Text
Liang, Z: Blow up rate for a porous medium equation with power nonlinearity. Nonlinear Anal.. 73, 3507–3512 (2010). Publisher Full Text
Zhou, S, Bai, X, Zheng, S: Large time behavior of solutions to a degenerate parabolic equation not in divergence form. J. Math. Anal. Appl.. 373, 252–263 (2011). Publisher Full Text
Xiang, Z, Chen, Q, Mu, C: Blow-up rate estimates for a system of reaction-diffusion equations with absorption. J. Korean Math. Soc.. 44, 779–786 (2007). Publisher Full Text