Abstract
By energy estimate approach and the method of upper and lower solutions, we give the conditions on the occurrence of the extinction and nonextinction behaviors of the solutions for a quasilinear parabolic equation with nonlinear source. Moreover, the decay estimates of the solutions are studied.
1. Introduction
The main goal of this article is to investigate the extinction behavior and decay estimate of the following parabolic initial boundary value problem Here, , , is an open bounded domain with smooth boundary , , , , and that are positive parameters, , and is a nonzero nonnegative function.
It is well known that this type of equation describes lots of phenomena in nature, such as heat transfer, chemical reactions, and population dynamics (one can see [1–4] for more detailed physical background). In particular, problem (1) can be used to describe the nonstationary flows in a porous medium of fluids with a power dependence of the tangential stress on the velocity of displacement under polytropic conditions. In this physical context, is the density of the fluid, denotes the momentum velocity, and stands for the nonlinear nonlocal source. The parameter acts as a characteristic of the medium, to be exact, the medium with is called Newtonian fluid, the medium with is called dilatant fluid, and that with is called pseudoplastic.
Extinction phenomenon, as one of the most remarkable properties that distinguish nonlinear parabolic problems from the linear ones, attracted extensive attentions of mathematicians in the past few decades (see [5–16] and the references therein). Especially, many authors devoted to concern with the extinction behavior of the following parabolic problem
Gu [17] discussed (2) with and , and concluded that the extinction phenomenon occurs if and only if . Tian and Mu [18] dealt with problem (2) with and , and derived that is the critical extinction exponent of problem (2). The authors of [19, 20] generalized the results in [18] to . The authors of [5, 21] concerned with the extinction behavior of problem (2) with and , and they pointed out that the effect of the nonlocal source term on the extinction behavior is very different from that of the local source . Recently, Zhou and Yang [22] dealt with the extinction singularity of problem (2) in the case and . For some relevant works on other types of nonlinear evolution equations, the readers can refer to the references [23–28].
However, to our best knowledge, there is no literature on the study of the extinction and decay estimate of the solutions for problem (1). Motivated by those works above, we consider the extinction property of problem (1). More precisely, our purpose is to understand how the nonlinear nonlocal source affects the extinction behavior of problem (1). In other words, the aim of this article is to evaluate the competition between the diffusion term which may produce extinction phenomenon and the nonlinear nonlocal source which may prevent the occurrence of the extinction phenomenon. We want to find a critical extinction exponent and give a complete classification on the extinction and nonextinction cases of the solutions to problem (1). Meanwhile, we will deal with the decay estimates of the extinction solutions.
Since equation (1) is degenerate (or singular) at the points where or , there is no classical solution in general, and hence we consider the nonnegative solution of (1) in some weak sense.
Definition 1. Let , and
We say that a function is a weak lower solution of problem (1) if holds for any and any nonnegative test function
Moreover,
Replacing by in the inequalities (4) and (6) leads to the definition of the weak upper solution of problem (1). We say that is a weak solution of problem (1) in if it is both a weak lower solution and a weak upper solution of problem (1) in .
Proposition 2. Assume that is a nonzero nonnegative function satisfying . Then, problem (1) has at least one local weak solution .
Remark 3. The proof of Proposition 2 is based on an approximation procedure and the Leray-Schauder fixed-point theorem, and it is standard and lengthy; so, we omit it here, while one can refer to the proof of Proposition 2.1 in [5] (or Proposition 2.3 in [19]) for more details. On the other hand, it is necessary to point out that the weak solution of problem (1) is unique for and . In the non-Lipschitz case or , the uniqueness of the weak solution seems to be unknown (See Remark 44.1 of §44.1 in [29]).
The main results of this article are stated as follows.
Theorem 4. Assume that . Then, the nonnegative weak solution of problem (1) vanishes in finite time provided that the nonnegative initial datum is sufficiently small. Moreover,
for , and
for , where , , , and are positive constants, given in Section 2.
Theorem 5. Assume that and are sufficiently large. Then, for any nonnegative initial datum , problem (1) admits at least one nonextinction weak solution.
Theorem 6. Assume that .
(1)The nonnegative weak solution of problem (1) vanishes in finite time provided that is sufficiently small. Moreover,for , and
for , where , , , and are positive constants, given in Section 2.
(2)Problem (1) admits at least one non-extinction weak solution for any nonnegative initial datum provided that is sufficiently large
2. Proofs of the Main Results
In this section, based on energy estimates approach and the method of upper and lower solutions, we will give the proofs of our main results.
Proof of Theorem 4. Multiplying equation (1) by and integrating over , one has where We now divide the proof into two cases according to the different values of .
Case 1. . For . It follows from Hölder inequality and (9) that
Using Hölder inequality and Sobolev embedding theorem, one has which is equivalent to
where is the embedding constant. Inserting (13) into (11) yields where
Now, if is sufficiently small satisfying then (14) leads to
By integration, one can deduce that which tells us that vanishes in finite time , where
For . By Sobolev embedding theorem, one obtains
Here, is the embedding constant. Combining (9) and (20), and in view of Hölder inequality, one arrives at where
Next, choosing sufficiently small such that then from (21), one has
Integrating (24) from to gives us that
which means that vanishes in finite time , where
Case 2. . If or , then the proof is the same as that in Case 1. We only need to focus our attention on the subcase and . Let be a bounded domain in satisfying . Denote be the first eigenvalue and be the corresponding eigenfunction of problem (One can see Lemma 2.3 of [18] for more details on the properties of the first eigenvalue and the corresponding eigenfunction of (27).)
We assume that . Put
Then, it is not difficult to show that is an upper solution of problem (1). Therefore, one has and
It follows from (9) and (29) that
For . It follows from (13) and (30) that where
Now, selecting sufficiently small satisfying then (31) tells us that
A simple integration of (34) over gives
which means that vanishes in finite time, where
For . Recalling (20) and (30), one obtains where
Next, if is sufficiently small such that then from (37), one arrives at
Integrating (40), one can claim that which tells us that vanishes in finite time, where
The proof of Theorem 4 is complete.
Proof of Theorem 5. Let be the first eigenvalue and be the corresponding eigenfunction of the following problem
In what follows, we assume that and . Define where . Then, it is easy to check that
In addition, one has
Define Then, one can verify that which implies that is a strict weak lower solution of problem (1) if
Now, consider the following problem
Using Leray-Schauder fixed-point theorem, we can prove that problem (47) admits at least one weak solution , and we know that by the weak maximum principle. In addition, the weak solution is also a weak upper solution of problem (1).
Up to now, we have constructed a pair of weak upper and lower solutions , . If , then problem (1) admits a weak solution satisfying . By the definitions of and , one has
Take , where is a monotone increasing smooth approximation of the following function
It is easy to check that as . Letting , it follows from (48) that where is a positive constant. Using Gronwall’s inequality, one can conclude that , a.e., in . Furthermore, since does not vanish, neither does . The proof of Theorem 5 is complete.
Proof of Theorem 6. (1)For . It follows from (14) that
If is sufficiently small such that , then above inequality tells us that
which means that vanishes in finite time , where
For . It follows from (21) that
If is sufficiently small such that , then (54) leads to
which implies that vanishes in finite time , where (2)Let
One can easily prove that is a weak nonextinction lower solution of problem (1) if . On the other hand, let be a weak solution of problem (47) with ; then, is a weak upper solution of problem (1). Similar to the process of proof of Theorem 5, one can claim that problem (1) has at least one nonextinction weak solution . The proof of Theorem 6 is complete.
3. Conclusion
In the present article, we mainly focus on the extinction phenomenon and the decay estimates of the solution to a quasilinear parabolic equation with a coupled nonlinear source. By analyzing the competition between the coupled nonlinear source term and the fast diffusion term, along with energy estimates approach and the method of upper and lower solutions, we show that is the critical extinction exponent of the solutions. That is, if , then for sufficiently small initial datum, the solution possesses extinction property, while if , then for any nonnegative initial datum, problem (1) admits at least one nonextinction solution provided that is sufficiently large. In the critical case , whether the solution vanishes or not depends on the size of the parameter .
Our next work is to study the numerical extinction phenomenon of the parabolic problems like (1). We hope to give some numerical examples for our theoretical researches in the near future.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
Conflict of interest statement is included without existing competing interests.
Acknowledgments
The first author is sincerely grateful to professor Chunlai Mu of Chongqing University for his encouragements and suggestions. The authors would like to thank the editor and anonymous referee for their careful reading and important comments and suggestions. This research is supported by the Natural Science Foundation of Hunan Province (Grant No. 2019JJ50160).