| 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) satisfiesThe 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 solutionTheorem2.(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<c1, c2<b2,1≤l=s,1≤p=q(which is a special case of lq=sp),0<α≤<s,0<β≤p and α,β <2. If λ1<c(?) min then all the solutions of (7) blow up in finite time.In the second part of Chapter2we investigate the global existence and blow-up properties of solutions to a class of weakly coupled degenerate parabolic systems with localized sources where x0∈Ω is a fixed point, a, b>0,(u0, v0) and (f, g) satisfy the following assumptions:(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)liminfsâ†'∞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.(â…¡) 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 followingTheorem5.(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<p, q <1) and the initial data (u0,u0) satisfy(A5) There exists a constant σ>0such that where σ, k1, k2>0depend only on a, b,p, q, we have the following blow-up rate estimates:Theorem6.(Blow-up rate) Suppose/(s)=sp,g(s)=sq,0<p,q <1,(u0,u0) satisfies (A1),(A2) and (A5). If(u,u) is a solution of (8) that blows up atT, then there exist constants C1-C4depending only on a, b, p, q such thatIn Chapter3we investigate the properties of solutions to some parabolic equation-s(systems) coupled with nonlocal boundary conditions, which can be viewed as perturba-tions of homogeneous Dirichlet boundary conditions. Unlike problems with homogeneous Dirichlet boundary conditions, the nonlocal boundary conditions also play an important role in determining whether the solutions will blow up in finite time or not. First we consider a porous medium equation with a localized source where m>1, a>0, f2is a bounded domain in RN (N≥1), with smooth boundary (?)Ω, k(x, y) is a nonnegative continuous function defined on (?)Ω×Ω, satisfying fΩ k(x, y)dy>0for all x∈(?)Ω, and uo(x) is a positive continuous function denned on Q, which satisfies the compatibility condition uo(x)=fΩk(x,y)u0(y)dy for x∈(?)(?)Ω. f(s) satisfies the assumptions as follows:(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 followingTheorem7. Assume that f satisfies (HI) and (H2).(â… ) 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.(â…¡) If fΩk(x,y)dy>1for any x∈(?)Ω, then the solutions to (9) blow up in finite time provided that(â…¢) 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 sâ†'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 <C such thatIn the second part of Chapter3, we apply the methods and skills used in the first part to deal with a class of porous medium systems with nonlocal boundary conditions Here m,n>1,(f,g) represents a nonlocal source or a localized source, i.e.(f(v),g(u))=(a fΩ vpdx, b fΩ uqdx) or (f(u),g(u))=(aup(x0,t),buq(x0,t)).(u0,u0) and (k1,k2) satisfy the following assumptions(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.(â… ) 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.(â…¡) 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.(â…¢) Suppose that fΩk1(x,y)dy,fΩk2{x,y)dy <1for any x∈(?)Ω.(i) If pq <mn, then every solution to (10) is global;(ii) Ifpq=mn, then the solutions to (10) exist globally provided that (a, b) is suitably small; Moreover, if p,q≥1, then the solutions to (10) blow up in finite time provided that (u0,u0) and (a,b) are appropriately large;(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 followingTheorem12. Assume that (H8)-(H10) hold.(â… ) If for any x∈(?)Ω, then the solutions to (11) blow up in finite time if and only if for some δ>0;(â…¡) 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.(â…¢) 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<p <1), that07>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 thatIn 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<m <1,6=0, in which case the equation in (12) is a porous medium equation coupled with a nonlocal source. Since the nonlinearity in (12) is non-Lipschitz when0<q <1, we can not establish a comparison principle for all the solutions. To overcome this difficulty, we first obtain a maximal solution of (12) by utilizing the method of regularization and a priori estimates, and then show that the maximal solution satisfies a sub-solution compar-ison principle. By using this sub-solution comparison principle we can prove, for a special case, the uniqueness of weak solutions to (12) as well as the existence of non-extinction solutions when the source is strong. We also prove that the solutions vanish in finite time and derive the decay estimates when the source is weak, by using the methods of integral estimates and Sobolev embedding theorems. The main results are the following:Theorem14.(â… ) Assume that q <m. Then for any nonnegative initial datum u0G L∞(Ω), the maximal solution U(x,t) of (12) can not vanish in finite time.(â…¡) Assume that q> m. Then every solution of (12) vanishes in finite time for small initial data u0G L∞(Ω).(â…¢) 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.(â… ) Assume that0<q <m(p-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∈G L∞(Ω)∩W01,p(Ω).(â…¡) 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.(â…¢) 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)ï¼›(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<p <2, b^0, in which case the equation in (12) is a p-Laplace equation coupled with both a nonlocal source and an absorption term. We will always assume that u0G L∞(Ω)∩W01,p(Ω) in this part. When the absorption term is linear (r=1),(12) can be changed into a equation without absorption terms under the transformation that v(x,t)=ebtu(x,t). However, when r≠1, the integral method used in the above two sections is not so powerful and it is much more complicated to determine whether the solutions will vanish in finite time or not. Only partial answers can be obtained by combining the famous Gagliardo-Nirenberg’s inequality with the methods of integral estimates. In order to draw a more clear picture of how the diffusion term, the nonlocal source term and the absorption term affect the extinction of the solutions to (12) and to show the critical extinction exponents, we first establish a weak comparison principle for our problem and then investigate whether the solution will vanish or not, by constructing vanishing super-solutions or positive sub-solutions. Our main results in this part are the followingTheorem16. Assume one of the following holds:(i)q>p-l.(ii) min{g,1}>r,or q=r<1with a|Ω|<b. Then the solutions of (12) vanish in finite time for small initial data.Theorem17. Assume q <p-1. If q <r, or q=r with b <αγ,then Problem (12) admits at least one non-extinction solution for any non-negative initial data. Here7>0is a constant depending only on 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 <r <1, then the solutions of (12) vanish in finite time for any non-negative initial data with an=1, and vanish in the sense of limtâ†'+∞||u(·,t)||2=0when a\Ω\≤λ1.(iii) If q <r with aμ>1, or q <1≤r with an=I, then Problem (12) admits at least one non-extinction solution for any positive initial data. |
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