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本文主要研究几类退化或奇异反应扩散方程(组)解的性质.所讨论的问题包括非局部边界条件,局部源,局部化源,非局部源以及吸收项对方程解的整体存在、爆破或熄灭的影响.本文共分为四章,主要内容如下：
第一章为绪论.我们首先概述本文所研究问题的实际背景和国内外相关的研究工作,然后阐述我们所要讨论的问题及所使用的方法.
在第二章中,我们研究具齐次Dirichlet边界条件的拟线性方程组解的整体存在与爆破性质.这种性质反映了方程中源项、吸收项和扩散项之间的平衡.我们首先考虑一类具内部吸收项的强耦合抛物方程组这里Ω是RN(N≥1)中具有光滑边界(?)Ω的有界区域,α,β≥0,ai,bi,ci(i=1,2)>0,L,s,p,q≥1.初值(u0,v0)满足
由于方程组是强耦合的,比较原理一般不再成立,这给我们证明解的整体存在和爆破带来一定的困难.又因为方程是退化的,所以解的局部存在性也不能由经典理论直接得到.此外,内部吸收项的存在也给我们判断解是否会在有限时刻爆破增加难度.我们利用正则化的方法结合先验估计技巧得到了问题(1)正古典解的局部存在性;接着我们再次利用先验估计技巧并借助常微分方程组的比较原理证明了一定条件下问题(1)解的整体存在性;最后我们克服强耦合和内部吸收项带来的困难,借助积分估计的技巧用Lp模逼近L∞模证明了一定条件下解的爆破性质.与以往人们研究强耦合问题不同的是,我们这里不需要假设解关于时间单调递增.我们的主要结果如下：
定理1.(局部存在性)假设初值(u0,v0)满足(A0).则问题(1)至少存在一个正古典解
定理2.(整体存在性)如果lq>sp,则问题(1)至少有一个有界的正整体解;如果lq=sp,则当b1pc2l>b2lc1p(即b1qc2s>b2sc1q)时,问题(1)至少存在一个正整体解.
定理3.(爆破)假设b1
(A2)u0(x)=v0(x)=0,(?)u0/(?)v<0,(?)v0/(?)v<0,x∈(?)Ω,这里v是(?)Ω上的单位外法向量;
(A3)f,g∈C([0,∞))nC1((0,∞)),f(0)=g(0)=0,且当x∈(0,∞)时,f(x),9(x)>0,f’(x),g’(x)>0；
同第一部分一样,在证明局部解的存在性时,我们必须借助正则化方法来克服边界退化所带来的困难.为了研究问题(2)解的整体存在性,我们利用它的结构构造了一个单个方程的初边值问题,并证明了这个单个方程解的整体存在性;然后从该单方程问题的解出发构造了问题(2)的整体存在的上解.为了给出问题(2)的解在有限时刻爆破的充分条件,我们首先构造了它的一个径向对称的下解,然后将问题转化为一个具局部源的方程组.这方面的结果如下：
定理4.(整体存在与爆破)(Ⅰ)如果ab≤1/θ2或者存在δ>0使得则问题(2)的解都是整体存在的.这里θ>0是只依赖于区域Ω和x0的常数.
(Ⅱ)如果ab>λ12(B),且存在δ>0使得则问题(2)的解在有限时刻爆破.这里B是以x0为心的包含于Ω的最大的球,λ1(B)>0是-△在B上的第一特征值.
当f还满足一定的增长条件时,我们证明了爆破解的爆破集是整个区域,即
定理5.(爆破集)如果问题(2)的解(u,v)在T时刻爆破,且存在δ>0使得则(u,v)是全局爆破的.
如果f(s)=sp,g(s)=sq(0
定理6.(爆破速率)假设f(s)=sp,g(s)=sq,0
(H1)f(s)∈C[0,∞)∩C1(0,∞)；
(H2)f(0)≥0,f’(s)>0,s∈(0,∞).
为了清晰刻画非局部边界条件、非线性扩散项和非线性源对解的长时间行为的综合影响,我们首先对问题建立比较原理;然后通过仔细分析各项对解的整体存在或爆破的作用并借助一些特殊构造的椭圆方程的解和常微分方程的解构造了恰当的上、下解,进而获得了问题(3)的解整体存在和爆破的充分条件.主要结果是：
定理7.假设{满足(H1)和(H2).
(Ⅰ)如果对任意x∈(?)Ω∫Ωk(x,y)dy=1,则问题(3)的解整体存在的充分必要条件是存在s0>0使得
(Ⅲ)如果对任意x∈(?)Ω有∫Ωk(x,y)dy>1,则(3)的解在有限时刻爆破的充分条件是存在s0>0使得
(Ⅲ)如果对任意x∈(?)Ω有∫Ωk(x,y)dy<1,则当以下两个条件中有一个成立时,(3)的解是整体存在的：
(ⅰ)存在s0>0使得
(ⅱ)当s→0+时,f(s)=o(s)且u0(x)适当小
(ⅲ)α适当小
当f满足一个比(H1)更强的条件时,我们可以对任意权函数k(x,y)给出(3)的解在有限时刻爆破的一个充分条件.
定理8.假设f满足(H2)和下面的条件
(H3)f(s)∈C1[0,∞),.f’(s)/sm1在(0,∞)上单调不减,且对某个s0>0有则对任意k(x,y),(3)的解对充分大的a或u0都在有限时刻爆破.
当边界权函数较小时,我们证明了爆破解的爆破集是整个区域.此外,当解关于时间单调递增时,我们还得到了特殊情形时解的爆破速率.
定理9.假设且u(x,t)是问题(3)在有限时刻T处爆破的解.则u(x,t)是全局爆破的;此外如果u0(x)还满足
(H4)u0(x)∈C2+“((?)),0<α<1,并且存在常数δ>δ1(a,m,p)>0使得则当f(s)=sp(p>1)时,存在正常数c
(H5)k1(x,y),k2(x,y)是定义在(?)Ω×(?)上的非负连续函数,且满足对任意x∈(?)Ω,∫Ωk1(x,y)dy,∫Ωk2(x,y)dy>0；
(H6)存在α∈(0,1)使得(u0,v0)∈[C2+α((?))]2；u0(x),v0(x)>0,x∈Ω；u0(x)=∫Ωk1(x,y)u0(y)dy,v0(x)=∫Ωk2(x,y)v0(y)dy,x∈(?)Ω.
类似于研究单个方程时的过程,我们首先对问题(4)建立弱比较原理,然后借助这个弱比较原理并通过构造恰当的上下解来给出问题(4)的解整体存在或爆破的充分条件,揭示非局部边界条件和非线性源项对爆破的影响.最后我们研究了边界权函数适当小时解的爆破速率.这里我们只叙述(∫,g)为非局部源时的结论.当(∫,g)为局部化源时,结论是类似的.我们的主要结果如下：
定理10.(Ⅰ)如果对任意x∈(?)∫Ω有Ωk1(x,y)dy=∫Ωk2(x,y)dy=1,则(4)的解整体存在的充要条件是pq≤1.
(Ⅱ)如果对任意x∈(?)Ω有∫Ωk1(x,y)dy,∫Ωk2(x,y)dy>1,则当pq>1时,(4)的解在有限时刻爆破.
(Ⅲ)删假设对任意x∈(?)Ω有∫Ωk1(x,y)dy,∫Ωk2(x,y)dy<1.
(ⅰ)如果pq
(ⅲ)如果pq>mn,则当(u0,v0)或(a,b)适当小时,(4)的解整体存在;如果还有p,q≥1,则当(u0,v0)足够大时,其解在有限时刻爆破.
当p,q≥1,∫Ωki(x,y)dy≤1(i=1,2)且(u,v)关于时间单调递增时,我们证得了问题(4)解的爆破速率
定理11.假设(u,v)是(4)在T时刻爆破的解,p,q>1,∫Ωki(x,y)dy≤1(i=1,2)且(u0,v0)满足如下假设
(H7)u0,v0∈C2+“((?)),0<α<1,且存在δ0>0使得则存在正常数C1-C4使得
在第三章最后,我们研究一类具非局部源和非局部边界条件的非散度型方程这里初值u0(x),扩散系数∫(s)和权函数k(x,y)满足如下假设：
(H8)f∈C([0,∞))∩C1((0,∞)),且满足f(0)≥0,.f’(s)>0,s∈(0,∞)；
(H9)k(x,y)是定义在(?)Ω×Ω上的非负连续函数,且对任意x∈(?)Ω,∫Ωk(x,y)dy>0；
(H10)对某个α∈(0,1),u0∈C2+α((?)),u0(x)>0,x∈Ω；u0(x)=∫Ωk(x,y)u0(y)dy,x∈(?)Ω.
由于方程是非散度型的,我们没有办法直接对其建立所需的比较原理.为了克服这个困难,我们首先建立一个极值原理,然后基于这个极值原理建立比较原理.进而我们根据边界扰动的不同来构造恰当的上、下解,给出(5)的解在有限时刻爆破的充要条件,从而完整刻画非局部边界条件、非局部源项和非线性扩散项对解的长时间行为的影响.同时,我们也就任意权函数k(x,y)得到了问题(5)解的一个整体上界.据我们所知,这种结果在以往研究具非局部边界条件的拟线性方程的文献中是从未得到过的.我们的主要结果是：
定理12.假设条件(Ⅱ8)-(Ⅱ10)圳成立.
(Ⅰ)如果对任意x∈(?)Ω有∫Ωk(x,y)dy≥1,则(5)的解在有限时刻爆破的充分必要条件是存在δ>0使得
(Ⅱ)如果对任意x∈(?)Ω有∫Ωk(x,y)dy<1,则问题(5)的解u(x,t)在有限时刻爆破的充要条件是aμ>1且这里μ>0是只依赖于区域Ω的常数.
(Ⅲ)如果存在δ>0使得那么对任意的k(x,y),只要a适当大,问题(5)的解在有限时刻爆破.
最后我们研究了解的爆破速率.与以往文献中研究具非局部边界条件问题解的爆破速率不同的是,我们不需要假设解关于时间t具有单调性.
定理13.假设∫(s)=sp(0
在第四章中,我们研究非线性方程解的另一种奇性——解的有限时刻熄灭.我们将分三节分别研究不同情形时如下快扩散方程解的熄灭性质,考察非线性扩散、非线性非局部源和非线性吸收项对问题解的熄灭的综合影响,并首次(据我们所知)得到具非局部源的多孔介质方程和非牛顿多方渗流方程的临界熄灭指标.在第四章中,我们始终假设a,q>0,Ω是RN(N≥1)中的有界光滑区域,初值u0(x)≥,(?)0.
我们首先考虑当p=2,0
(Ⅲ)假设q=m.
(ⅰ)如果aμ>1,则对任意非负初值u0∈L∞(Ω),问题(6)的最大解U(x,t)不可能在有限时刻熄灭;
(ⅱ)如果aμ=1,则对任意光滑的正初值u0∈L∞(Ω),问题(6)的最大解U(x,t)不可能在有限时刻熄灭;
(ⅲ)如果aμ<1,则对任意非负初值u0∈L∞(Ω),问题(6)的惟一解u(x,t)在有限时刻熄灭.这里μ>0是只依赖区域(Ω的常数.
接着我们将上一节中得到的结果进行推广,考虑当m>0,p>1,m(p-1)<1,b=0时问题(6)解的熄灭性质.此时(6)中的方程是一个具非局部源项的非牛顿多方渗流方程.同第二节遇到的困难类似,我们也只能对某些特殊的上下解进行比较.借助积分估计、Sobolev嵌入定理和单调迭代技术我们可以证明q=m(p-1)是熄灭的临界指标.具体结果如下：
定理15.(Ⅰ)假设0
(Ⅲ)假设q=m(p-1).
(ⅰ)如果aK>,则对任意满足u0m∈L∞(Ω)∩w01，p(Ω)的非负初值u0,问题(6)至少存在一个不熄灭的弱解;
(ⅱ)如果aκ=1,则对任何满足u0m∈L∞(Ω)∩W01，p(Ω)的光滑正初值u0(x),问题(6)至少存在一个不熄灭的弱解;
(ⅲ)如果aκ<1,则对任意满足u0m∈L∞(Ω)∩W01,p(Ω)的非负初值u0,问题(6)至少存在一个熄灭弱解.这里κ>0是只依赖于区域(Ω的常数.
在第四章的最后,我们研究当m=1,1
(ⅰ)q>p-1.
(ⅱ)min{q,1}>r,或q=r<1且a|Ω|
定理18.假设q=p-1.
(ⅰ)如果aκ<1,则对任意非负初值,问题(6)的解都在有限时刻熄灭.
(ⅱ)如果q
Nonlinear difusion equations, as an important class of partial difusion equations,come from a variety of difusion phenomena appeared widely in nature. They arise frommany felds such as physics, chemistry, dynamics of biological groups, economics andfnance ect. In the past few decades, a large number of famous mathematicians bothin China and aboard have devoted themselves to these felds, and remarkable progresshas been achieved on the local existence and uniqueness of solutions, regularity, globalexistence, blow-up and extinction as well as estimates on the blow-up and extinctiontimes and rates, which enrich enormously both the theories and the contents of partialdiferential equations. Until now, the study of nonlinear difusion equations is still a veryactive research area.
In this paper we mainly investigate the properties of solutions to some classes ofdegenerate or singular reaction-difusion equations (systems). The topics include theefect of nonlocal boundary conditions, local sources, localized sources, nonlocal sources,absorption terms and the coupling among them on the global existence, blow-up andextinction properties of the solutions. This paper is divided into four chapters.
In Chapter1we frst describe the background of the problems considered in thispaper and recall briefly the related works obtained by the mathematicians both in Chinaand aboard. Then we state our problems and some methods and techniques that we shall use.
In Chapter2we study the global existence and blow-up of solutions to some quasi-linear parabolic systems coupled with homogeneous Dirichlet boundary conditions. First we consider a class of strongly coupled and degenerate quasi-linear parabolic system with inner absorption terms where Ω is a bounded domain in RN(N≥1) with smooth boundary (?)Ω,α and β are nonnegative constants,ai,bi,ci are positive constants and l,s,p,q≥1. The initial data (u0, v0) satisfies
The comparison principles do not hold in general for such problems since the equa-tions are strongly coupled, which brings great difficulties to us when proving the global existence and blow-up properties of solutions. The degeneracy on the boundary excludes the possibility of applying of the classical methods directly to the proof of the local ex-istence of classical solutions. Moreover, the existence of inner absorptions also makes it hard to give some sufficient conditions for the solutions to blow up in finite time. To overcome these difficulties, we first utilize the standard method of regularization as well as a priori estimates to prove the local existence of classical solutions; then we show that (7) admits at least one global solution by combining the a priori estimates with the comparison principles for ODEs;finally we overcome the difficulties brought by the absorption terms and the strongly coupled of the two exponents u and v, by using the method of integral estimates to approximate the L∞norms by Lp norms, and show that all the solutions of (7) blow up in finite time under certain conditions. Unlike most previ-ous works concerning similar problems, we do not need the assumption that the solutions are monotone increasing in t. Our results are as follows:
Theorem1.(Local existence) Assume that (u0,v0) satisfies (A0). Then (7) ad-mits at least one local classical solution
Theorem2.(Global existence) Iflq> sp, then Problem (7) admits at least one positive bounded global solution; If lq=sp, then (7) has at least one positive bounded global solution provided that b1pc2l> b2lc1p,i.e.,b1qc2s> b2sc1q.
Theorem3.(Blow-up) Suppose b1
(A1) u0(x),v0(x)∈C2+α(Ω)∩C(Ω) for some0<α <1, u0(x),v0(x)>0inΩ;
(A2) u0(x)=v0(x)=0,(?)u0/(?)v <0,(?)u0/(?)v<0on (?)Ω, where v is the outward normal vector on (?)Ω;
(A3) f,g∈C([0,∞))∩C1((0,∞)),f(0)=g(0)=0, and f,g>0,f',g'>0in (0,∞);
(A4)liminfs→∞f(s)/g(s)>0.
As in the first part, we encounter the same difficulties brought by the degeneracy on the boundary when proving the local existence of classical solutions, which force us to utilize the method of regularization. To study the global existence of solutions, we first introduce a scalar initial-boundary problem whose positive solution is global, and then construct a global super-solution of (8) by using the solution of the scalar problem. To give some sufficient conditions for the solutions to blow up in finite time, we first construct a radially symmetric sub-solution of (8) and then show that this radially symmetric sub-solution is also a super-solution of a system with local reaction terms whose solutions blow up in finite time. The main results read as follows:
Theorem4.(Global existence and blow-up)(I) If ab <1/θ2or for some δ>0, then the solution of (8) exists globally, where9>0is a constant depending only on Ω and x0.
(Ⅱ) If ab> λ12(B) and <∞for some δ>0, then the solution of (8) blows up in finite time. Here B is the biggest ball centered at x0and contained in Ω and λ1(B)>0is the first eigenvalue of—Δ in B.
We also show that the bow-up set of (u, v) is the whole domain when/satisfies certain growth conditions at infinity. The result in this direction is the following
Theorem5.(Blow-up set) If the solution (u,v) of (8) blows up in finite time and for some5, then (u, v) blows up globally.
When/(s)=sp,g(s)=sq(0
Theorem6.(Blow-up rate) Suppose/(s)=sp,g(s)=sq,0
(H1) f(s)∈C[0,∞)∩C1(0,∞);
(H2)/(0)>0and f'(s)>0in (0,∞).
To draw a clear picture of how the nonlocal boundary conditions, the nonlinear diffusion terms and the nonlinear source terms affect the long time behaviors of solutions and to give some sufficient conditions for the solutions of (9) to blow up in finite time or to exist globally, we first establish a comparison principle for (9) and then construct some suitable super and sub-solutions by using the solutions of some special elliptic equations and the solutions of some ODEs. The main result is the following
Theorem7. Assume that f satisfies (HI) and (H2).
(Ⅰ) If fΩ k(x,y)dy=1for any x∈(?)Ω, then the solutions to (9) exist globally if and only if for some s0>0.
(Ⅱ) If fΩk(x,y)dy>1for any x∈(?)Ω, then the solutions to (9) blow up in finite time provided that
(Ⅲ) Assume that fΩk(x,y)dy <1for all x∈(?)Ω. Then the solutions to Problem (9) are global if one of the following conditions holds:
(i) There exists a constant S>0such that
(ii) f(s)=o(s) as s→0+and uo(x) is sufficiently small;
(iii) a is suitably small.
We can give a sufficient condition for the solutions of (9) to blow up in finite time for any weight function k(x,y) when f satisfies an assumption stronger than (H1).
Theorem8. Assume that f satisfies (H2) and the following
(H3) f{s)∈C1[0,∞]; f(s)/sm-1is nondecreasing in (0,∞) and for some s0>0. Then, for any k(x,y), the solutions to Problem (9) blow up in finite time if a or u0(x) is sufficiently large.
When the weight function k(x,y) is in some sense small, we show that the blow-up set is the whole domain. Moreover, we obtain the blow-up rate estimates in the case that the solutions are increasing in t.
Theorem9. Assume and u(x,t) is a solution of (9) that blows up at T. Then u(x,t) blows up globally; Moreover, if u0(x) satisfies the following
(H4) u0(x) E C2+α(Ω) for some0<α <1and there exists a constant δ> δ1>0such that then for the case f(s)=sp(p>1), there exist two positive constants c
(H5) k1(x,y),k2(x,y) are continuous non-negative functions denned on (?)Ω×Ω and fΩ k1(x, y)dy,fΩ k2(x, y)dy>0for any x∈(?)Ω;
(H6)(u0,u0)∈[C2+α(Ω)]2for some α∈(0,1); u0(x),uo(x)>0, x∈Ω; u0(x)=fΩk1(x,y)u0(y)dy, u0(x)=fΩk2(x,y)uo(y)dy, x∈(?)Ω
Similar to the process that we study the scalar problem, we first establish a weak comparison principle suitable for (10), and then prove the global existence and blow-up results by constructing some proper super and sub-solutions, using the solutions of some special elliptic equations and ODEs. Finally, we derive the blow-up rate estimates when the weight functions are in some sense small. We only state the results when (f(u),g(u))=(a fΩ updx, b fΩuqdx), since the results of the other case are almost the same.
Theorem10.(Ⅰ) If fΩk1(x,y)dy=fΩk2(x,y)dy=1for any x∈(?)Ω, then the solutions to (10) exist globally if and only if pq <1.
(Ⅱ) If fΩ k1(x, y)dy, fΩk2(x,y)dy>1for any x∈(?)Ω, then the solutions to (10) blow up in finite time when pq>1.
(Ⅲ) Suppose that fΩk1(x,y)dy,fΩk2{x,y)dy <1for any x∈(?)Ω.
(i) If pq
(iii) If pq> mn, then the solutions to (10) exist globally provided that (a,b) or (u0,u0) is suitably small; Moreover, if p,q≥1, then the solutions blow up in finite time for sufficiently large (u0,u0).
When p, q≥1, and (u,uv) is increasing in t, we obtain the blow-up rate of (u,u).
Theorem11. Suppose that (u,v) is a solution to(10) which blows up at T, that p, q>1and that Assume also that (u0,u0) satisfies the following
(H7) u0,u0∈C2+α(Ω),0<α <1, and there exists a constant δ0>0such that Then there exist four positive constants C1-C4such that
In the last part of Chapter3we consider a quasi-linear parabolic equation not in divergence form coupled with nonlocal source and nonlocal boundary condition Here a is a positive constant,Ω is a bounded domain in RN with smooth boundary (?)Ω u0(x), f(s) and the weight function k(x,y) satisfy
(H8) f∈C([0,∞))∩C1((0,∞)) such that f(0)≥0, f'(s)>0, s∈(0,∞).
(H9) k(x, y) is continuous and nonnegative denned on (?)Ω×Ω satisfying fΩ k(x, y)dy>0for all x∈(?)Ω.
(H10) u0∈C2+α(Ω) for some α∈(0,1), u0(x)>0in Ω and u0(x)=fΩ k(x, y)u0(y)dy on (?)Ω.
Since the equation in (11) is not in divergence form, we can not establish a compari-son principle directly as we have done in the above two parts. To overcome this difficulty, we first prove a minimum principle, based on which we establish a comparison principle. Then by constructing suitable super-solutions and sub-solutions we give a complete clas-sification of f(s) and k(x, y) for the solution to blow up in finite time or not. Meanwhile, we obtain a uniform upper bound of solutions to (11) for arbitrary k(x,y). To the best of our knowledge, this kind of results have never been obtained in the previous works dealing with quasi-linear parabolic equations with nonlocal boundary conditions. The main results are the following
Theorem12. Assume that (H8)-(H10) hold.
(Ⅰ) If for any x∈(?)Ω, then the solutions to (11) blow up in finite time if and only if for some δ>0;
(Ⅱ) If fΩ k(x, y)dy <1for any x∈(?)Ω, then the solutions to (11) blow up in finite time if and only if aμ>1and for some δ>0; Here μ>0is a constant depending only on Ω.
(Ⅲ) If there exists a constant S>0such that then for any k(x,y), the solutions to (11) blow up in finite time provided that αγ>1. Here7G (0,μ) is a constant that only depends on Ω.
Finally we derive the blow-up rate by using the method of integral estimates. It is worth mentioning that unlike most previous works, we do not need the assumption that the solutions are monotone increasing in t.
Theorem13. Assume that/(s)=sp (0
1(7is the constant given in Theorem12), that (H9),(H10) hold and that fΩk(x,y)dy <1on (?)Ω. If the solution u(x,t) of (11) blows up at T, then there exist positive constants Ci (i=1,2,3) and q>1such that
Theorem16. Assume one of the following holds:
In Chapter4we consider another singular property of nonlinear parabolic equa-tions, the finite time extinction of solutions. We will devote ourselves to the extinction properties of solutions of the following fast diffusion problem and investigate how the nonlinear diffusion terms, the nonlinear source terms and the absorption terms affect the extinction of solutions to (12). We obtain for the first time(to our best knowledge) the critical extinction exponents for fast diffusive porous medium equations and polytropic nitration equations with nonlocal sources. Throughout Chapter4, we will always assume that a, q>0,Ω is a bounded smooth domain in RN(N≥1) and u0(x)≥,(?)0.
We first investigate the extinction properties of solutions to (12) when p=2,0
(Ⅲ) Assume that q=m.
(i) If αμ,>1, then for any nonnegative initial datum u0∈L∞(Ω), the maximal solution U(x,t) of (12) can not vanish in finite time;
(ii) If αμ=1, then for any positive smooth initial datum u0∈L∞(Ω), the maximal solution U(x,t) of (12) can not vanish in finite time;
(iii) If αμ <1, then for any nonnegative initial datum u0∈L∞(Ω), the unique solution of (12) vanishes in finite time. Here y>0is a constant depending only on Ω.
Next we generalize the results obtained in Section2to the case of m>0,p>1,m(p-1)<1,6=0, when the equation in (12) becomes a Non-Newtonian polytropic fiitration equation with a nonlocal source. Similar to the case in Section2, we can only prove the comparison principle for some special sub and super-solutions. By using integral methods, Sobolev embedding theorems and monotone iteration methods we prove that the critical extinction exponent of (12) is q=m(p-1). The main result reads as follows.
Theorem15.(Ⅰ) Assume that0 (Ⅱ) Assume that q> m(p-1) and u0m(x)∈L∞(Ω)∩W01,p(Ω). Then every non-negative weak solution of Problem (12) vanishes in finite time for appropriately small initial data u0.
(Ⅲ) Assume that q=m(p-1).
(i) If ak>1, then Problem (12) admits at least one non-negative and non-extinction weak solution u(x,t) for any non-negative initial datum u0with u0m∈L∞(Ω)∩W01,p(Ω);
(ii) If ak=1, then Problem (12) admits at least one non-negative and non-extinction weak solution u(x, t) for any smooth positive smooth initial datum u0with u0m G L∞(Ω)∩W01,p(Ω)；
(iii) If an <1, then Problem (12) admits at least one solution which vanishes in finite time for any non-negative initial datum u0with u0mG L∞(Ω)∩W01,P(Ω). Here κ>0is a constant depending only on Ω.
In the last part of Chapter4we study Problem (12) when m=1,1
(i)q>p-l.
(ii) min{g,1}>r,or q=r<1with a|Ω|
Theorem18. Assume q=p—1.
(i) If ak <1, then the solutions of (12) vanish in finite time for any non-negative initial data.
(ii) If q
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