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

Authors: Dengming Liu; Luo Yang
Journal: Advances in Mathematical Physics
 (2021)

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## 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.


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## Body

## 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(1)ut=divuα∇um−1∇u+λup∫Ωuqdx,x,t∈Ω×0,+∞,ux,t=0,x,t∈∂Ω×0,+∞,ux,0=u0x,x∈Ω¯.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 u0m+α/m∈L∞Ω∩W01,m+1Ω 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, ux,t is the density of the fluid, uα∇um−1∇u denotes the momentum velocity, and λup∫Ωuqdx 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
(2)ut−divax,t,u,∇φu=fx,t,u,x,t∈Ω×0,+∞,ux,t=0,x,t∈∂Ω×0,+∞,ux,0=u0x,x∈Ω¯.Gu [17] discussed (2) with ax,t,u,∇φu=∇u and fx,t,u=−up, and concluded that the extinction phenomenon occurs if and only if p∈0,1. Tian and Mu [18] dealt with problem (2) with ax,t,u,∇φu=∇up−2∇u and fx,t,u=λup, and derived that q=p−1 is the critical extinction exponent of problem (2). The authors of [19, 20] generalized the results in [18] to ax,t,u,∇φu=∇ump−2∇um. The authors of [5, 21] concerned with the extinction behavior of problem (2) with ax,t,u,∇φu=∇ump−2∇um and fx,t,u=λ∫Ωuqdx, and they pointed out that the effect of the nonlocal source term λ∫Ωuqdx on the extinction behavior is very different from that of the local source λuq. Recently, Zhou and Yang [22] dealt with the extinction singularity of problem (2) in the case ax,t,u,∇φu=∇um and fx,t,u=λup∫Ωuqdx. 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 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.Definition 1.
LetΣT=Ω×0,T, and
(3)S=u∈L2pΣT∩L2qΣT∩L2ΣT;u∈C0,T;L1Ω;∇um+αm∈Lm+1ΣT.We say that a functionux,t∈S is a weak lower solution of problem (1) if
(4)∫Ωux,Tζx,Tdx+∬ΣTuα∇um−1∇u·∇ζ−uζtdxdt≤∫Ωux,0ζx,0dx+∬ΣTλup∫Ωuqdxζdxdtholds for any T>0 and any nonnegative test function
(5)ζ∈u∈L2ΣT;u∈C0,T;L2Ω;ut∈L2ΣT;∇u∈Lm+1ΣT;u∂Ω=0.Moreover,(6)ux,0≤u0xforx∈Ω¯,andux,t≤0forx,t∈∂Ω×0,T.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 ΣT if it is both a weak lower solution and a weak upper solution of problem (1) in ΣT.Proposition 2.
Assume thatu0x is a nonzero nonnegative function satisfying u0m+α/m∈L∞Ω∩W01,m+1Ω. Then, problem (1) has at least one local weak solution ux,t∈S.Remark 3.
The proof of Proposition2 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.Theorem 4.
Assume that0<m+α<p+q. Then, the nonnegative weak solution of problem (1) vanishes in finite time provided that the nonnegative initial datum u0x is sufficiently small. Moreover,
(7)u2m+αm≤u02m+αm1−d4t11−m−α,t∈0,T1,u2m+αm≡0,t∈T1,+∞,for mN−m−1/Nm+m+1−1≤α<1, and uN1−m−α/m+1≤u0N1−m−α/m+11−d8t1/1−m+α,t∈0,T2,uN1−m−α/m+1≡0,t∈T2,+∞,
for−m<α<mN−m−1/Nm+m+1−1, where T1=d4−1, T2=d8−1, d4, and d8 are positive constants, given in Section 2.Theorem 5.
Assume that0<p+q<m+α<1 and λ are sufficiently large. Then, for any nonnegative initial datum u0x, problem (1) admits at least one nonextinction weak solution.Theorem 6.
Assume that0<m+α=p+q<1.
(1)
The nonnegative weak solution of problem (1) vanishes in finite time provided that λ is sufficiently small. Moreover,(8)u2m+αm≤u02m+αm1−d14t11−m−α,t∈0,T3,u2m+αm≡0,t∈T3,+∞,
formN−m−1/Nm+m+1−1≤α<1, and uN1−m−α/m+1≤u0N1−m−α/m+11−d15t1/1−m−α,t∈0,T4,uN1−m−α/m+1≡0,t∈T4,+∞,
for−m<α<mN−m−1/Nm+m+1−1, where T3=d14−1, T4=d15−1, d14, and d15 are positive constants, given in Section 2.
(2)
Problem (1) admits at least one non-extinction weak solution for any nonnegative initial datum u0x provided that λ 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 us and integrating over Ω, one has
(9)1s+1ddt∫Ωus+1dx+sm+1m+α+sm+1∫Ω∇um+α+sm+1m+1dx=λ∫Ωup+sdx∫Ωuqdx,where
(10)s=m+αm,ifmN−m−1Nm+m+1−1≤α<1,N1−m−α−m−1m+1,if−m<α<mN−m−1Nm+m+1−1.
We now divide the proof into two cases according to the different values ofp+q.Case 1.
m+α<p+q≤1. For mN−m−1/Nm+m+1−1≤α<1. It follows from Hölder inequality and (9) that
(11)m2m+αddt∫Ωu2m+αmdx+mm+αm∫Ω∇um+αmm+1dx≤λΩ2−mp+q+m+α2m+α∫Ωu2m+αmdxmp+q+m+α2m+α.Using Hölder inequality and Sobolev embedding theorem, one has(12)∫Ωu2m+αmdx≤Ω1−2m+αN−m−1Nm+1m+α∫Ωum+αm·Nm+1N−m−1dx2m+αm+α·N−m−1Nm+1≤κ1Ω1−2m+αN−m−1Nm+1m+α∫Ω∇um+αmm+1dx2m+αm+1m+α,which is equivalent to
(13)κ1−m+1m+α2m+αΩ1−m+1N−m+1m+α2m+α∫Ωu2m+αmdxm+1m+α2m+α≤∫Ω∇um+αmm+1dx,whereκ1=κ1α,m,N is the embedding constant. Inserting (13) into (11) yields
(14)ddt∫Ωu2m+αmdx+d1∫Ωu2m+αmdxm+1m+α2m+α≤d2∫Ωu2m+αmdxmp+q+m+α2m+α,where
(15)d1=2m+αmm−1m+α−mκ1−m+1m+α2m+αΩ1−m+1N−m+1m+α2m+α,d2=λ2m+αm−1Ω2−mp+q+m+α2m+α.Now, ifu0x is sufficiently small satisfying
(16)d3=d1−d2∫Ωu02m+αmdxmp+q−m−α2m+α>0,then (14) leads to
(17)ddt∫Ωu2m+αmdx+d3∫Ωu2m+αmdxm+1m+α2m+α≤0.By integration, one can deduce that(18)u2m+αm≤u02m+αm1−d4t+11−m−α,which tells us that ux,t vanishes in finite time T1=d4−1, where
(19)d4=md31−m−α2m+α−1u02m+αmm+α−1.For−m<α<mN−m−1/Nm+m+1−1. By Sobolev embedding theorem, one obtains
(20)∫Ωus+1dxm+α+sm+1s+1=∫ΩuNα+m+sN−m−1dxN−m−1Nm+1≤κ2∫Ω∇uα+m+sm+1m+1dx1m+1.Here,κ2=κ2α,m,N is the embedding constant. Combining (9) and (20), and in view of Hölder inequality, one arrives at
(21)ddt∫Ωus+1dx+d5∫Ωus+1dxm+α+ss+1≤d6∫Ωus+1dxp+q+ss+1,where
(22)d5=ss+1m+1κ2m+α+s−1m+1,d6=λs+1Ω2−p+q+ss+1.Next, choosingu0x sufficiently small such that
(23)d7=d5−d6∫Ωu0s+1dxp+q−m−αs+1>0,then from (21), one has
(24)ddt∫Ωus+1dx+d7∫Ωus+1dxm+α+ss+1≤0.Integrating (24) from 0 to t gives us that
(25)uN1−m−αm+1≤u0N1−m−αm+11−d8t+11−m−α,which means thatux,t vanishes in finite time T2=d8−1, where
(26)d8=d7m+11−m−αN1−m−α−1u0N1−m−αm+1m+α−1.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. Let Ω~ be a bounded domain in RN satisfying Ω⊂⊂Ω~. Denote λ~1 be the first eigenvalue and Ψ~x 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).)
(27)−divUα∇Um−1∇U=λUα+1Um−1,x∈Ω~,Ux=0,x∈∂Ω~.We assume thatmaxx∈Ω~Ψ~x=1. Put
(28)U1x,t=μΨ~xwithμ∈maxx∈Ω¯u0xΨ~x,minx∈Ω¯λ~1Ψ~m+αxλΩ1p+q−m−α.Then, it is not difficult to show thatU1x,t is an upper solution of problem (1). Therefore, one has ux,t≤μΨ~x≤μ and
(29)λ∫Ωup+sdx∫Ωuqdx≤λΩμp+q−1∫Ωus+1dx.It follows from (9) and (29) that
(30)1s+1ddt∫Ωus+1dx+sm+1m+α+sm+1∫Ω∇um+α+sm+1m+1dx≤λΩμp+q−1∫Ωus+1dx.FormN−m−1/Nm+m+1−1≤α<1. It follows from (13) and (30) that
(31)ddt∫Ωu2m+αmdx+d1∫Ωu2m+αmdxm+1m+α2m+α≤d9∫Ωu2m+αmdx,where
(32)d9=λΩ2m+αm−1μp+q−1.Now, selectingu0x sufficiently small satisfying
(33)d10=d1−d9∫Ωu02m+αmdxm1−m−α2m+α>0,then (31) tells us that
(34)ddt∫Ωu2m+αmdx+d10∫Ωu2m+αmdxm+1m+α2m+α≤0.A simple integration of (34) over 0,t gives
(35)u2m+αm≤u02m+αm1−d4t+11−m−α,which means thatux,t vanishes in finite time, where
(36)d4=md101−m−α2m+α−1u02m+αmm+α−1.For−m<α<mN−m−1/Nm+m+1−1. Recalling (20) and (30), one obtains
(37)ddt∫Ωus+1dx+d5∫Ωus+1dxm+α+ss+1≤d11∫Ωus+1dx,where
(38)d11=λs+1Ωμp+q−1.Next, ifu0x is sufficiently small such that
(39)d12=d5−d11∫Ωu0s+1dx1−m−αs+1>0,then from (37), one arrives at
(40)ddt∫Ωus+1dx+d12∫Ωus+1dxm+α+ss+1≤0.Integrating (40), one can claim that
(41)uN1−m−αm+1≤u0N1−m−αm+11−d8t+11−m−α,which tells us that ux,t vanishes in finite time, where
(42)d8=d12m+1N−1u0N1−m−αm+1m+α−1.The proof of Theorem4 is complete.Proof of Theorem 5.
Letλ1 be the first eigenvalue and Ψx be the corresponding eigenfunction of the following problem
(43)−divUα∇Um−1∇U=λUα+1Um−1,x∈Ω,Ux=0,x∈∂Ω.In what follows, we assume thatΨx>0 and maxx∈ΩΨx=1. Define ft=1−e−ct1/1−p−q, where c∈0,1−p−qλ∥Ψ∥qq−λ1. Then, it is easy to check that
(44)f0=0,andft∈0,1fort>0.In addition, one has(45)f′t+λ1fm+αt−λ∥Ψ∥qqfp+qt≤0.DefineU2x,t=ftΨx. Then, one can verify that
(46)U2t−divU2α∇U2m−1∇U2−λU2p∫ΩU2qdx=f′tΨx+λ1fm+αtΨm+αx−λ∥Ψ∥qqfp+qtΨpx<f′t+λ1fm+αt−λ∥Ψ∥qqfp+qtΨpx≤0,which implies that U2x,t is a strict weak lower solution of problem (1) if λ>λ1∥Ψ∥q−q.Now, consider the following problem(47)ut=divuα∇um−1∇u+λu++1p∫Ωu++1qdx,x,t∈Ω×0,∞,ux,t=0,x,t∈∂Ω×0,∞,ux,0=u0x≥0,x∈Ω¯.Using Leray-Schauder fixed-point theorem, we can prove that problem (47) admits at least one weak solution U3x,t, and we know that U3x,t≥0 by the weak maximum principle. In addition, the weak solution U3x,t is also a weak upper solution of problem (1).Up to now, we have constructed a pair of weak upper and lower solutionsU3x,t, U2x,t. If U2x,t≤U3x,t, then problem (1) admits a weak solution u~ satisfying U2≤u~≤U3. By the definitions of U2 and U3, one has
(48)∫ΩU2x,t−U3x,tζx,tdx−∫ΩU2x,0−U3x,0ζx,0dx+∬ΣtU2α∇U2m−1∇U2−U3α∇U3m−1∇U3·∇ζdxdτ−∬ΣtU2−U3ζτdxdτ≤λ∬ΣtU2p∫ΩU2qdx−U3++1p∫ΩU3++1qdxζdxdτ=λ∬ΣtU2p∫ΩU2q−U3++1qdx+U2p−U3++1p∫ΩU3++1qdxζdxdτ.Takeζx,t=HεU2m+α/m−U3m+α/m, where Hεr is a monotone increasing smooth approximation of the following function
(49)Hr=1,r>0,0,otherwise.It is easy to check thatHε′r→δr as ε→0. Letting ε→0, it follows from (48) that
(50)∫ΩU2−U3+dx≤λ∬ΣtU2p∫ΩU2q−U3++1qdxHU2m+αm−U3m+αmdxdτ+λ∬ΣtU2p−U3++1p∫ΩU3++1qdxHU2m+αm−U3m+αmdxdτ≤d13∬ΣtU2−U3+dxdτ,where d13 is a positive constant. Using Gronwall’s inequality, one can conclude that U2x,t≤U3x,t, a.e., in Ω×0,∞. Furthermore, since U2 does not vanish, neither does u~. The proof of Theorem 5 is complete.Proof of Theorem6.
(1)
FormN−m−1/Nm+m+1−1≤α<1. It follows from (14) that(51)ddt∫Ωu2m+αmdx≤d2−d1∫Ωu2m+αmdxm+1m+α2m+α.Ifλ is sufficiently small such that d1−d2≥0, then above inequality tells us that
(52)u2m+αm≤u02m+αm1−d14t+11−m−α,which means thatux,t vanishes in finite time T3=d14−1, where
(53)d14=md1−d21−m−α2m+α−1u02m+αmm+α−1.For−m<α<mN−m−1/Nm+m+1−1. It follows from (21) that
(54)ddt∫ΩuN1−m−αm+1dx≤d6−d5∫ΩuN1−m−αm+1dxN−m−1N.Ifλ is sufficiently small such that d5−d6≥0, then (54) leads to
(55)uN1−m−αm+1≤u0N1−m−αm+11−d15t+11−m−α,which implies thatux,t vanishes in finite time T4=d15−1, where
(56)d15=m+1d5−d6N−1u02m+αmm+α−1.(2)
Let(57)U4x,t=1−p−qλ∥Ψ∥qq−λ1t11−p−qΨx.One can easily prove thatU4x,t is a weak nonextinction lower solution of problem (1) if λ>λ1∥Ψ∥q−q. On the other hand, let U5x,t be a weak solution of problem (47) with p+q=m+α; then, U5x,t 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 u~. 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 thatp+q=m+α is the critical extinction exponent of the solutions. That is, if m+α<p+q, then for sufficiently small initial datum, the solution possesses extinction property, while if p+q<m+α, then for any nonnegative initial datum, problem (1) admits at least one nonextinction solution provided that λ is sufficiently large. In the critical case p+q=m+α, whether the solution vanishes or not depends on the size of the parameter λ.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.



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