Extinction Phenomenon and Decay Estimate for a Quasilinear Parabolic Equation with a Nonlinear Source

Here, Ω ⊂ RN , N ≥m + 1, is an open bounded domain with smooth boundary ∂Ω,m, p, q, and λ that are positive parameters, 0 <m + α < 1, and um+α/m 0 ∈ L∞ðΩÞ ∩W1,m+1 0 ðΩÞ 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, uðx, tÞ is the density of the fluid, uαj∇ujm−1∇u denotes the momentum velocity, and λup Ð Ω uqdx stands for the nonlinear nonlocal source. The parameter m acts as a characteristic of the medium, to be exact, the medium with m = 1 is called Newtonian fluid, the medium with m > 1 is called dilatant fluid, and that with 0 <m < 1 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


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, Ω ⊂ R N , N ≥ m + 1, is an open bounded domain with smooth boundary ∂Ω, m, p, q, and λ that are positive parameters, 0 < m + α < 1, and u m+α/m 0 ∈ L ∞ ðΩÞ ∩ W 1,m+1 0 ðΩÞ 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][2][3][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, uðx, tÞ is the density of the fluid, u α j∇uj m−1 ∇u denotes the momentum velocity, and λu p Ð Ω u q dx stands for the nonlinear nonlocal source. The parameter m acts as a characteristic of the medium, to be exact, the medium with m = 1 is called Newtonian fluid, the medium with m > 1 is called dilatant fluid, and that with 0 < m < 1 is called pseudoplastic.
The authors of [5,21] concerned with the extinction behavior of problem (2) with aðx, t, u,∇φðuÞÞ = j∇u m j p−2 ∇u m and f ðx , t, uÞ = λ Ð Ω u q dx, and they pointed out that the effect of the nonlocal source term λ Ð Ω u q dx on the extinction behavior is very different from that of the local source λu q . Recently, Zhou and Yang [22] dealt with the extinction singularity of problem (2) in the case aðx, t, u,∇φðuÞÞ = ∇u m and f ðx, t, uÞ = λu p Ð Ω u q dx. For some relevant works on other types of nonlinear evolution equations, the readers can refer to the references [23][24][25][26][27][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 u = 0 or ∇u = 0, there is no classical solution in general, and hence we consider the nonnegative solution of (1) in some weak sense.
We say that a function uðx, tÞ ∈ S is a weak lower solu- holds for any T > 0 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 u is a weak solution of problem (1) in Σ T if it is both a weak lower solution and a weak upper solution of problem (1) in Σ T .

Proposition 2.
Assume that u 0 ðxÞ is a nonzero nonnegative function satisfying u m+α/m 0 ∈ L ∞ ðΩÞ ∩ W 1,m+1 0 ðΩÞ. Then, problem (1) has at least one local weak solution uðx, tÞ ∈ S. 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 p ≥ 1 and q ≥ 1. In the non-Lipschitz case 0 < p < 1 or 0 < q < 1, 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.
(2) Problem (1) admits at least one non-extinction weak solution for any nonnegative initial datum u 0 ðxÞ provided that λ is sufficiently large

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 u s and integrating over Ω, one has We now divide the proof into two cases according to the different values of p + q. Case 1. m + α < p + q ≤ 1. For mðN − m − 1/Nm + m + 1 − 1Þ ≤ α < 1. It follows from Hölder inequality and (9) that Using Hölder inequality and Sobolev embedding theorem, one has which is equivalent to where κ 1 = κ 1 ðα, m, NÞ is the embedding constant. Inserting (13) into (11) yields d dt where Now, if u 0 ðxÞ is sufficiently small satisfying By integration, one can deduce that which tells us that uðx, tÞ vanishes in finite time For −m < α < mðN − m − 1/Nm + m + 1 − 1Þ. By Sobolev embedding theorem, one obtains Here, κ 2 = κ 2 ðα, m, NÞ is the embedding constant. Combining (9) and (20), and in view of Hölder inequality, one arrives at d dt 3

Advances in Mathematical Physics
where Next, choosing u 0 ðxÞ sufficiently small such that then from (21), one has d dt Integrating (24) from 0 to t gives us that which means that uðx, tÞ vanishes in finite time Case 2. m + α < 1 < p + q. If p < 1 or q < s + 1, then the proof is the same as that in Case 1. We only need to focus our attention on the subcase p ≥ 1 and q ≥ s + 1. LetΩ be a bounded domain in R N satisfying Ω ⊂ ⊂Ω. Denoteλ 1 be the first eigenvalue andΨðxÞ 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 max x∈ΩΨ ðxÞ = 1. Put Then, it is not difficult to show that U 1 ðx, tÞ is an upper solution of problem (1). Therefore, one has uðx, tÞ ≤ μΨðxÞ ≤ μ and It follows from (9) and (29) that For mðN − m − 1/Nm + m + 1 − 1Þ ≤ α < 1. It follows from (13) and (30) that d dt where Now, selecting u 0 ðxÞ sufficiently small satisfying Next, if u 0 ðxÞ is sufficiently small such that which tells us that uðx, tÞ vanishes in finite time, where The proof of Theorem 4 is complete.