Existence And Blow-up Of Solutions To Reaction-Diffusion Equations (Systems) With Nonlinear Sources

In the real world, reaction-difusion phenomena exists widely. Themathematical models are partly reduced to the study of some parabolicequations or systems. In fluid mechanics, chemistry, theory of phase tran-sitions, image processing, biological populations, as well as areas such aspercolation theory etc, have raised a number of parabolic equations withappropriate initial and boundary conditions to describe the difusion phe-nomena, In the recent decades, many scholars have made significant progressin studying such models.As we known, although people use some linear equations to describesome difusion models, but most of the real models should only be de-scribed with nonlinear equations. Therefore, the parabolic equations usedto describe these models usually have nonlinear terms, and also may bedegenerate or singular. Although the nonlinear problems may accuratelyreflect some actual phenomena, but they also cause some difculties in work-ing out these problems. For instance, the nonlinear terms (difusion terms,source terms, boundary terms) will play a role in promoting or hindering inblowing-up phenomena of the solutions.This paper mainly studies some reaction-difusion problems with non- linear sources. The topics include the reaction-diffusion terms with p(x)-Laplace, nonlocal boundary conditions, local sources, localized sources, non-local sources and coupling among them in the critical exponents of solutions. This thesis consist of three chapters.In the Introduction, we first recall the background of the reaction-diffusion equations and summarize the main results of the present thesis and some problems to be studied.In Chapter1, we study the existence and blow-up phenomenon for some diffusion equations with nonlocal boundary conditions. There are different degrees of the complexity of the nonlinear properties for the local sources and localized sources. We have overcome the difficulties caused by the nonlinear terms. With the use of the method of super-sub solution and comparison principle, we prove the existence and blow-up of the solutions.In the first part of Chapter1, we consider the following porous medium equations with localized sources and nonlocal boundary conditions:where m>1, a>0, n>0, p≥0and q>0are constants and Ω is a bounded domain in RN (N≥1)with smooth boundary (?Ω), x0∈Ω is a fixed point. k(x,y)≠0is a nonnegative continuous function for x∈(?)Ω and y y∈Ω, while the initial datum u0(x) is a positive continuous function. It is in general difficult to study the system, we regularize the problem (1), prove some estimates for the solutions of the regularized problem, and hence obtain the local existence of solutions. Then, we prove a weak compari-son principle and obtain some results on the global existence and blow-up property of solutions by constructing a supersolution(subsolution) of the systems. Our main results are the following:Theorem1Assume that∫Ωk(x,y)dy=1, for all x∈(?)Ω. Then every solution to (1) exists globally for p+q≤1and0<n≤1, while it blows up in finite time for p+q>1and n=1.Theorem2Assume that∫Ωk(x,y)dy>1, for all x∈(?)Ω. The solution to (1) blows up in finite time provided that p+q>1, n≥1and uo(x) is large enough.Theorem3Assume that∫Ωk{x,y)dy<1, for all x∈(?)Ω.(ⅰ) If p+q<m and0<n≤1, then every solution to (1) exists globally;(ⅱ) If p+q=m and0<n≤1, then every solution to (1) exists globally if a is sufficiently small;(ⅲ) If p+q>m and n≥1, then the solution to (1) exists globally provided that uo(x) or a is small, while u(x,t) blows up in finite time if uo(x) is large enough.Theorem4Assume that p+q≥m. Then the solution to (1) blows up in finite time provided that uo(x) and a is large enough.In the second part of Chapter1, we discuss the porous medium equa-tions with nonlocalized sources and nonlocal boundary conditions:where m>1, a>0, n>0, p≥O and q>0are constants and Ω is a bounded domain in M.N (N≥1)with smooth boundary (?)Ω, x0∈Ω is a fixed point. k(x,y)≠0is a nonnegative continuous function for x∈(?Ω) and y∈Ω, while the initial datum u0(x) is a positive continuous function. We also prove the existence and blow-up of solutions. The main results are the following: Theorem5Assume that∫Ωk(x,y)dy=1,for all x∈(?)Ω. Then every solution to (2) exists globally for p+q≤1and0<n≤1, while it blows up in finite time for p+q>1and n=1.Theorem6Assume that J^k(x,y)dy<1, for all x∈(?)Ω.(ⅰ) If p+g<m and0<n≥1, then every solution to (2) exists globally;(ⅱ) If p+q=m and0<n≥1, then every solution to (2) exists globally if a is sufficiently small;(ⅲ) If p+q>m and n≥1, then the solution to (2) exists globally provided that uq(x) or a is small, while u(x,t) blows up in finite time if u0(x) is large enough.In the third part of Chapter1, we also consider the following nonlinear parabolic equation with nonlocal boundary conditions and local sources:where0<m<1,a>0,n>0are constants and Ω is a bounded domain in M.N (N≥l1)with smooth boundary (?)Ω. k(x,y)≠0is a nonnegative continuous function for x∈(?)Ω and y∈Ω, while the initial datum u0(x) is a positive continuous function. Assume that:(A1) f∈C([0,∞))∩C1(0,∞)) such that∫(0)≥0and f'(s)>0,s∈(0,oo).(A2) f is convex in (0,∞), and∫s0∞ds/f(s)<∞, for some s0≥0.The main results are the following:Theorem7Assume that the assumptions (Al) hold. Then every solution to (3) exists globally if a is small enough or if n≥1, f(s)=o(s) as s→0and uo(x) is sufficiently small. Theorem8Assume that the assumptions (A1)-(A2) hold, then the positive solution u(x,t) of Problem (3) blows up in finite time provided that a is sufficiently large or uo(x) is large enough.In Chapter2, on the base of the first Chapter with constant exponent, we discuss firstly the existence and blow-up of solutions of diffusion equa-tions for nonlocal boundary condition with variable exponent. We also study the initial boundary problems for diffusion equation systems with variable exponent. In the second chapter, for the source term and complexity of the boundary conditions, we conducted a more detailed discussion and finally obtain some results to the existence and blow-up of the solutions.In the first part of Chapter2, we consider the following diffusion equa-tion with variable exponent and nonlocal boundary conditions:where Ω RN is a bounded domain in RN (N≥1)with smooth boundary (?)Ω, p>0and l0. c(x,t)≥0is a Holder continuous with x∈Ω and t≥0, and k(x,y,t)≥0is continuous function for all x∈(?)Ω, y∈Ω and t≥0, while the initial datum uo(x) is a positive continuous function. We have overcome the difficulties caused by the nonlocal sources and the boundary conditions with variable exponents. Then we discuss the existence and blow-up of solutions of the problem. The main results are the following theorems:Theorem9Let Ω∈RN be a bounded smooth domain and let max{p-,l}>1, with p(x) satisfying given conditions. Then, for a sufficiently large initial datum u0(x), the positive solution u of Problem (4) blows up in a finite time. Then,in the second part of Chapter2,we discuss the following the initial boundary value problem for the diffusion systems with variable ex-ponents:where α>0,β>0are constants,Ω∈RN is a bounded domain with a smooth boundary (?)Ω and0<T<∞,QT=Ω×[0,T),ST denotes the lateral boundary of the cylinder QT.The source terms are of the form f(u)=up1(x)and g(u)=up2(x),where the exponents p1(x),p2(x):Ω→(1,+∞)satisfy the following con-ditions:1<p2=inf x∈Ωp2(x)≤p2(x)≤p2+=sup x∈Ωp2(x)<+∞,(6)1<p2=inf x∈Ωp2(x)≤p2(x)≤p2+=sup x∈Ωp2(x)<+∞.(7)With this type of source terms,we study the existence and blow-up of the solutions.The main results are the following theorems:Theorem10Let Ω∈RN be a bounded smooth domain,p1(x),p2(x) satisfy the given conditions(6)and(7),and assume that u0(x)and u0(x)are nonnegative,continuous and bounded.Then there exists a T0,0<T0≤∞, such that Problem(5)has a nonnegative and bounded solution(u,v)in QT0.Theorem11Let Ω∈RN be a bounded smooth domain and(u,v) a positive solution of Problem(5),with p1(x),p2(x)satisfying the given conditions. Then,for sufficiently large initial datum(u0(x),u0(x)),there exists a finite time T*>0such that sup0≤t≤T*|||(u,v)|||=+∞.Finally,in the third part of Chapter2,we investigate the existence and blow-up of solutions to the nonlinear parabolic systems with variable expo-nents: where Ω∈RN is a bounded domain with smooth boundary (?)Ω and0<T<∞,QT=Ω×[0,T),ST denotes the lateral boundary of the cylinder QT,the exponents p1(x),p2(x)are given functions satisfying the following conditions.The source terms are of the form f1(u,v)=a1(x)up1(x) adn f2(u,v)=a2(x)up2(x), or respectively. And where the continuous functions α1(x), α2(x):Ω→R satisfy:0<c1≤α1(x)≤C1<+∞,0<c2≤a2(x)<C2<+∞.(10)The main results are the following theorems:Theorem12Let Ω∈RN be a bounded smooth domain and let (u,v) be a positive solution of Problem (8), with p1(x), p2(x), α1(x), α2(x) satisfying given conditions (6),(7) and (10). Then any solutions of Problem (8) are blow up at finite time T*, if the initial datum (u0(x), vo(x)) satisfieswhere ψ is the first eigenfunction on Ω for some fixed positive constant C which will depend only on the domain Ω and the bounds C1, C2given condition.Theorem13Let (u,v)∈C2×C2be a solution of problem (9), and the given conditions (6),(7) and (10). Then there exist sufficiently large initial data u0, u0, u1, u1such that any solutions of Problem (9) are blow up at finite time T*.In Chapter3, we consider the existence and blow-up properties of solu-tions for two classes of high-order diffusion equations with variable expo-nents. The substantial difficulties of the problem are caused by the general comparison principle being invalid.In the first part of Chapter3, we investigate the following high-order diffusion equations to initial boundary problem: where the exponents p(x), q(x)∈C(Ω) satisfy the following conditions: We obtain:Theorem14Assume that u0(x)∈H01(Ω)∩H2(Ω), and p(x)≥q(x) satisfy the given condition (12). Then, there exists at least one weak solution to initial boundary Problem (11).Theorem15Assume that p(x)<q(x)≤p-+1satisfy the given condi-tions (12). Then, there exists at least one weak solution to initial boundary Problem (11).In the second part of Chapter3, we study the following high-order pseudo-parabolic equations with the following initial boundary values problem:where ΩRN is a bounded domain with a C1,1boundary (?)Ω, QT=Ω×(0, T], ST denotes the lateral boundary of the cylinder QT, μ>0is the viscosity coefficient and α>0is a parameter, the term μ(?)u/(?)t in (1.1) is interpreted as due to viscous relaxation effects. The exponents p, q:Ω→(1,+∞) are continuous functions. The main results are the following theorems:Theorem16Assume that u∈E W and p(x)≥q(x) satisfy the given conditions (12). Then, there exists at least one weak solution to Problem (13). Theorem17Let u0(x)∈H01(Q)∩H2(Ω),and p(x)≥q(x)≥max{1,(2N)/(N+1)} satisfy the given conditions(12).Then the initial-boundary value problem (9) has at least one solution.Theorem18Assume that1<q(x)<(Np(x))/(N-p(x))satisfy the given conditions (12).Then,there exists at least one weak solution for Problem(13).