Study Of Solutions To Some Degenerate Or Singular Diffusion Equations (Systems)

Difusion equation is an important class of partial diferential equations. A varietyof phenomena that arise from many fields such as physics, chemistry, economics anddynamics of biological groups in nature, which can be modeled by difusion equation. Inrecent years, more and more mathematicians, physicists, chemists and biologists in Chinaand aboard focused on difusion equations and many important problems are developed.The studying of global existence, blow up, extinction, critical exponent, the lower boundof blow-up time, the time and rate of extinction has been an important aspect in the filedof difusion equations. In this paper we mainly investigate the blow up of solutions fora class of nonlinear parabolic equations with positive initial energy, the global existenceand blow up of solutions for double degenerate parabolic equations and the extinctionand decay estimate of solutions for a class of fast difusive nonlinear parabolic equations.This paper is divided into four chapters.In Chapter1we mainly introduce the background of the problems considered in thispaper and recall briefly related works has obtained by the mathematicians. Then westate our problems we shall discuss and the methods and techniques we shall use.In Chapter2we study the blow up of solutions for a class of nonlinear parabol-ic equations with positive initial energy. First, we consider a class of porous medium equation where f2is a bounded domain of RN, N≥3with a smooth boundary (?)Ω, m>1, u0{x) is a nonnegative function and u0∈L∞(Ω)∩W-1,0p(Ω), f(u) is a C(R) function satisfying the following conditions: where B is the optimal constant of embedding inequality.The Problem (8) doesn’t always have classical solutions and the local existence can’t be derived since the above equations are degenerate. First, we utilize the standard method of regularization as well as priori estimate for the above problem to prove the local existence of classical solution; then we prove that the weak solutions of Problem (8) blow up in finite time when the initial energy is positive. The previous work proved that the solution blew up by the comparison principle and the method of the super-solution and sub-solution. In the general, the comparison principle doesn’t hold and the method of constructing super-solution and sub-solution is complex. We prove the solutions of Problem (8) blow up in finite time by constructing the energy functional methods. We introduce the following energy function Main results are following:Theorem1.(Local existence) Assume that h(s) C1(R), the condition (Al) holds, f(s) E C(R) and satisfy the following conditions h(s)>0,|ms-1f(s)|≤h(sm),there is a T’∈(0,T),Problem (8) has a then for any initial u0∈L∞(Ω)∩W1,0p(Ω), there is a T’(0,T), Problem (8) has a weak solution u that satisfies um∈L∞(Ω×(0,T’))∩L2((0,T’)):H10(Ω)),(um|1/2)l∈L2(Ω×(0,T’)) Theorem2.(Blow-up)Assume that N>2,2<r≤2N/N-2,f(s) satisfies (A1) and smf(s)≥rF(s)≥｜s｜mr；furthermore,if u0m≥0and E(0)<E1,then the solulion of Problem (8) blows up in finite time,where E1=(1/2-1/r)B-2r/r-2.In the second part of Chapter2we investigate the semi-linear parabolic equations when the initial energy is positive,the solution of Problem (9) blows up, where Ω (?) RN(N≥3) is bounded,(?)Ω is Lipschitz continuous and u0(x)≥0.p(x) satisfies the following conditions:However,due to the existence of variable exponents and the lack of homogeneity, we found that the previous method failed in solving our problem.To overcome these difficulties,we have to look for new methods or techniques to discuss some properties of solutions of the above problem. We prove that the solution of nonlinear parabolic equations blows up when initial energy is positive by the relationship between the mode and norm.The energy function is following: E(t)=∫Ω [1/2|▽u(x,t)|2-1/p(x)+1up(x)+1(x,t)]dxThe main results are following:Theorem3.(Blow-up)Assume that p(x) satisfies the condition (A2) and the fol-lowing conditions hold: then the solution of Problem (9) blows up in finite time,where OZl satisfie and B is the optimal embedding constant.Due to the technology we used,we can’t discuss that the solution of Problem(9) blows up or not,when1<p≤(?)2p|-1and initial energy is positive.In the three part of Chapter2,we investigate the following porous medium equation where Ω(?)RN(N≥3)is boundcd,the boundary (?)Ω is Lipschitz continuous,u0≥0, and m>0. Duc to the existeuce of variable exponent and quasi-linear item△um,we need to reconstruct the monotone decreasing energy function according to the equation of couerete form. There is a gap between mode and norm and the control function is not differential in the variable exponent space. We need to costruct a new piecewise function and transformed into the problem of ordinary differential. We give the lower bound of blow-up time by the method of auxiliary function and Sobolev inequality.We introduce the energy function E(t)=∫Ω[1/2|▽um(x,t)|2-m/m+p(x)um|p(x)]dx We can derive that the following results to Problem(10): Theorem4.(Blow-up) Assume thal initial E(0)<E1,||▽um0||22>α1,p(x) salisfies(A2),and m<(?)≤p(.)≤p+≤m(N+2/N-2)(N≥3),then the solulion of ProblemTheorem5.(Lower bounds for blow-up time)Assume that u(x,t) is the non-negative weak solution of Problem (10),Ω(?)Rn(n≥3) is bounded.We define where κ is defined as follows κ> max{2(n-2)(p+-1)-(m-1)n, m+1}.If the solution of Problem (10) blows up in finite time T, the the lower bound of time T satisfies where κ1, κ2, m1, m2and ε are constants. κ1, κ2, m1, m2, m3; m4and ε are defined as followswhen m=1,the Problem (10) is the Problem (9). We extend the range of exponent in source from (?)2p+-1<p-<p+<N+2/N-2to1<p-≤p(.)<p+≤N+2/N-2.The hot sources can promote the blow up of the solutions and the cold sources can hinder the blow up of the solutions, we discuss that the critical exponent of blow up for a class of doubly degenerate parabolic equations with hot sources and cold sources in the Chapter3. We discuss the following parabolic equations: where p, q>2, m, n>1, r1,r2, S1, s2>1, α,β>0, Ω is a bounded domain of RN, N≥1with a smooth boundary (?)Ω. The initial data uo(x), vo(x) satisfy compatibility and the following conditions:Problem(11)should be the synthesis of Newton porous systems and Non-newton porous systems.We give the critical exponent of Problem(11)by the mcthod of construct-ing sub-solution and super-solution. The difficulty is that constructing of sub-solution and super-solution should be comprehensive.In view of inner absorption,we need to con-struet a new ordinary differential equation in order to construet the blow-up sub-solution. Let μ=max{m(p一1),s1|,κ=max(n(q-1),s2}.The main results are following:Theorem6.Assume that r1r2<μκ,then the nonncgative solution of Problem (11) exisls globally.Theorem7.Assume that r1r2<μκ,then the nonnegative solution of Problem(11) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small inilial values.Theorem8.Assume that r1r2=μκ,(?)(x)and (?)(x) satisfy the following equations:-div(|▽(?)m|p-2▽(?)m)=1,x∈Ω；(?)(x)=1, x∈(?)Ω, and-div(|▽(?)n|q2▽(?)n)=1,x∈Ω；(?)(x)=1,x∈(?)Ω.Then we have the following conclusions:(α)Suppose that s1>m(p-1) and s2>n(q-1),if αr2βs1≥|Ω|r2+s1,then∫Ω(?)r2dx>βΨs2,then the solution of Problem (11) blows in finile time for larg initial the solution of Problem (11)exists globally for small initial values;if∫ΩΨr1dx>αΨs1, values,(b) Suppose that s1<m(p-1) and s2<n(q-1),if (∫Ω(?)r2dx)1/r2.(∫ΩΨdx)1/m<p-1)≤1, then the solution of Problem (11) exists globally for small initial valves;if∫ΩΨr1dx>1, ∫Ωφr2dx>1,then the solution of Problem(11) blows up in finite time for large initial values.(c)Suppose that s1<m(p-1) and s2>n(q-1),if∫Ωφr2dx≤β|Ω|-s2/r1,then the solution of Problem (11) exists globally; if∫Ωψr1dx>1,∫Ωφr2dx>αψs2,then the solution of Problem(11) blows up in finite time for large initial values.(d) Suppose that s1>m(p-1) and s2<n(q-1),if∫Ωψr1dx≤α|Ω|-s1/r2,then the solution of Problem (11) exists globally;if∫Ωψr1dx>αφs1,∫Ωφr2dx<1,then the solution of Problem (11) blows up in finite time for large initial values.In the Chapter4,we consider the extinction of solution for a class of nonlinear parabolic equation with nonlinear absorptions.The absorptions are cold sources,which contribute the extinction of solution, and we give the sufficient conditions about the extinction and decay estimates of solution by applying LP model estimate methods and interpolation inequalities.First,we consider the following porous equation with nonlinear absorptions where0<m<1,p,λ,β>0,Ω(?) RN is bounded and the boundary (?)Ω is smooth.We assume that0<u0(x)∈L∞(Ω)∩W1,20(Ω), λ1is the first eigenvalue of the following equations whereφ1(x)≥0and||φ1||L∞(Ω)=1are the eigenfunction corresponding to the first eigenvalue λ1.The main results are following:Theorem9.Assume that0<m=p<1,λ1is the first eigenvalue of Problem(13), we have(1)if N-4/N≤m<1and λ<4mλ1/(m+1)2,then the weak solution of Problem (12) vanishes in finite time and the decay estimate is following (2)If0<m<N-4/N, then the weak solution of Problem (12) vanishes and the decay estimate is following then initial u0is small enough or λ is small enough, the non-negative weak solution of Problem (12) vanishes in finite time.In the second part of Chapter4, we investigate the extinction and decay estimates of solutions for the p-Laplacian equations with Nonlinear absorptions and nonlocal sources. We discuss the following equation where1<p<2, k,q, λ>0,0<r<1, Ω(?)RN,(N≥2) is a bounded domain with smooth boundary and u0(x)∈L∞(Ω)∩W1,p0(Ω) is a non-negative function.We will discuss Problem (14) by applying the extinction and decay estimate of solu-tions for ordinary differential equation, the LP model estimate methods and Gagliardo-Nirenberg interpolation inequalities. The main results are following:Theorem11. Assume that p-1=q with r<1, then the non-negative nontrivial weak solution of Problem (14) vanishes in finite time for any non-negative initial data provided that|Ω|or λ is sufficiently small, (1)For the case2N/N+2≤p<2,we have(2)For the case1<p<2N/N+2,we haveTheorem12.Assume that r<1,(1)If2N/N+2≤p<2with q>κ1-1=2rp+N(p-1-r)/2p+N(p-1-r),then the non-negative nontrivial weak solution of Problem (14) vanishes in finite time provided that u0(or|Ω|) or λ) is sffficiently small and(2)If1<p<2N/N+2with q>k2-s=(s+1)rp+N(p-1-r)/(s+1)p+N(p-1-r),then the non-negati(?) nontrivial weak solution of Problem (14) vanishes if finite time provided that u0(or|Ω|or λ)is sufficiently small and where s and k2are defined as the above.M4=kC2-k2(N,p,r,s)/C(ε2)-λ|Ω|2+s－q/1+s||u0||q+s-κ21+s>0 and T4=||u0||1+s-k21+s/M4(s+1-κ2).