Nonlinear Diffusion Equations And Systems With Periodic Sources

In this monograph, we consider nonlinear diffusion equations and systems withperiodic sources. We mainly study the non-Newtonian polytropic filtration equationwith logistic periodic sources, the weakly coupled non-Newtonian filtration systemwith periodic sources and the weakly coupled non-Newtonian polytropic filtrationsystem with periodic sources. In recent years, the study on nonlinear diffusionequations and systems with periodic sources are noticed extensively. It becausesthat this kind of equations can be used to model a great many problems whichcome from many fields such as physics, chemistry, biology and have vivid practicalapplied background. Furthermore, the discussion on nonlinear diffusion equationsand systems with periodic sources brings forward a challenge to mathematics, hence,the study on this kind equation has profound theory value.Diffusion equations, as an important class of parabolic equations, come from avariety of diffusion phenomena appeared widely in nature. They are suggested asmathematical models of physical problems in many fields such as filtration, phasetransition, image segmentation, biochemistry and dynamics of biological groups. Inthis monograph, we investigate a kind of important problems of nonlinear diffu- sion equations, that is the periodic problem. A great many states and processes innature transform regularly and periodic diffusion equations are important mathematicsmodels to incarnate this phenomena. The study on periodic diffusion equationsbrings forward many significative problems, the essential problems includethe eixstence, the stability of periodic solutions and periodic principal eigenvalues.In the seventieth age of last century, the study on periodic problems of diffusionequations catched many internal and overseas mathematicians' eyes and had manycontents and results. The equations in these classical works have nonlinear periodicsources and are linearly diffused. The disscusion on periodic equations withnonlinear diffusion began in recent ten years. Typical works come down to nonlineardiffusion equations with strongly nonlinear sources, weakly nonlinear sources ornonlocal terms.Let us review some results on diffusion equations with periodic sources. Firstin the case of classical Logistic equationwhere a(x, t), b(x, t) are continuous functions and of periodic with respect to t. ShairAhmad and Alan C. Lazer discussed the existence of nontrivial nonnegative periodicsolutions and long behavior of solutions of the initial boundary value problem. Alongwith the development of Logistic equation more deeply, in some domains, mathematiciansintroduced some degenerate parabolic equations. Particularly, Hess, Pozioand Tesei used the monotonicity method to deal with the equationwhere m > 1 and a(x, t) is a continuous function and of periodic in t. They discussedthe existence of nontrivial nonnegative periodic solutions, multiplicity andsupports pro and so on. Mizoguchi applied the Leray-Schauder fixed point theory to investigate the equations with superlinear forcing termswhere m > 1, h(t) is a positive periodic function and f satisfies some structureconditions which are fulfilled automatically by f = u^p with p > m. The equation(1) can incarnate practicality more factually. The termâ–³u^m models a tendency toavoid crowding. Yin, Wang discussed the following porous medium equation withlogistic periodic sourceswhere m > 1, 0≤a≤m, a(x, t),b(x, t) are positive periodic smooth functions. Bymeans of Leray-Schauder fixed point theory, they obtained the existence of nontrivialnonnegative periodic solutions. With the extensive study on periodic parabolicequations, many authors dealted with the periodic problem for a certain kind ofquasilinear parabolic equations with degeneration, which include as their importantexamples, the following non-Newtonian filtration equationwhere p≥2 and the function f is periodic in t. In particular, when f(x, t, u) =g(x, t), Seidman disscussed the Dirichlet periodic boundary value problem by usingof functional method. His results were improved by Nakao for the following equationwhere f,g are periodic functions,β'(u) > 0 when u≠0 and satisfies some constructconditions. The authors established the existence of periodic solutions underthe conditions that f(x,t,u)u≤C|u| with C≥0. Recently, for periodic problemsof non-Newtonian filtration equation with nonlinear sources or nonlocal terms, many researchers obtained the existence of nontrivial nonnegative periodic solutions.With the development of periodic problems of Newtonian filtration equationand non-Newtonian filtration equation, many researchers began the discussion onnon-Newtonian polytropic filtration equation. Especially, Gao, Wang considered thefollowing periodic boundary value problem for doubly degenerate parabolic equationwith nonlocal termswhere m>1, p > 2, a = a(x, t) is periodic in t. By using the homotopy invarianceof Leray-Schauder's degree, they obtained the existence of nontrivial nonnegativeperiodic solutions.Along with the researches on single nonlinear periodic diffusion equations moredeeply, mathematicians began the discussion on nonlinear periodic diffusion systems.In particular, Yin, Wang disscussed the reaction diffusion system with weaklycoupled periodic sourceswhere b₁(x, t),b₂(x, t) are nonnegative functions and periodic in t. By the method ofsupersolution and subsolution, the author obtain the existence of nontrivial nonnegativeperiodic solutions for the Dirichlet periodic boundary value problem, and alsodisscussed the asymptotic behavior of nonnegative solutions of the initial boundaryvalue problem. At the same time, many mathematicians investigated the non-Newtonian filtration system where p₁,p₂ > 2,α₁,β₁,α₂,β₂ are nonnegative constants. The results include someestimates near blow-up points for radially symmetric positive solutions of (6), thenonexistence of positive solutions of the related elliptic systems. However, there isno result on the non-Newtonian filtration system with periodic sources.In this thesis, we mainly consider the non-Newtonian polytropic filtration equation,the non-Newtonian filtration system and the non-Newtonian polytropic filtrationsystem with periodic sources in three chapters respectively.In the first chapter, we discuss the initial boundary value problemwhere m > 1, p > 2, 1≤α< m(p-1),β> 0,Ω(?) R^N is a bounded domain withsmooth boundary (?)Ω, a = a(x,t),b = b(x,t) are nonnegative continuous functionsand of T-periodic with respect to t, u₀ is a nonnegative bounded smooth functionand satisfies the compatibility condition u₀^m(x)∈W₀^1,p(Ω). Our interest lies in theexistence of nontrivial nonnegative periodic solutions of the problem (7)-(8) andasymptotic bounds of nonnegative solutions of the problem (7)-(9).Noticing that the equation (7) includes the Newtonian filtration equation (p =2) and the non-Newtonian filtration equation (m = 1) formally, So the method forit should be synthetic. In fact, we can use the methods for the above two equationsto deal with it. In this chapter, we utilize Gronwall's inequality to establish thecomparison principle. Then by the first eigenvalue and its corresponding eigenfunctionto the p-Laplacian operator on some domains, with respect to homogeneousDirichlet data, respectively, we construct a pair of well-ordered positive supersolutionand subsolution. By monotonicity and the property of the Poincare map, we obtain the existence of nontrivial nonnegative periodic solutions of the problem (7)-(8). In order to discuss asymptotic bounds of nonnegative solutions of the problem(7)-(9), we make use of Moser iteration technique to obtain a priori upper boundof nonnegative periodic solutions of (7)-(8), by which we obtain the existence ofthe maximum periodic solution and asymptotic bounds of nonnegative solutions of(7)-(9). We also prove that the support of the periodic solution is independent oftime.In the second chapter, we discuss the following weakly coupled non-Newtonianfiltration systemwhere p₁,p₂ > 2,α₂,β₁ > 0,α₁,β₂≥0, b₁(x,t), b₂(x,t) are positive continuousfunctions and periodic in t,Ω(?) R^N is a bounded domain with smooth boundary(?)Ω. Firstly, we obtain the global existence of weak solutions for the initialboundary value problem by generalized method. Then we establish the comparisonprinciple by Gronwall's inequality. By the eigenvalue problem for non-Newtonianfiltration equation and system, we construct a pair of well-ordered positive supersolutionand subsolution. Then we establish the existence of nontrivial nonnegativeperiodic solutions of the periodic boundary value problem for (10) by the method ofsupersolutiona and subsolution. At last, the maximum dorm estimate is obtained bymeans of energy estimates and De Giorgi iteration technique. By which we obtainasymptotic bounds of nonnegative solutions of the initial boundary value problem.In the third chapter, we study the weakly coupled non-Newtonian polytropicfiltration system where m₁,m₂ > 1, p₁,p₂ > 2,α₂,β₁ > 0,α₁,β₂≥0, b₁(x,t), b₂(x,t) are positivecontinuous functions and periodic in t,Ω(?) R^N is a bounded domain with smoothboundary (?)Ω. Since the system (11) includes Newtonian filtration system and non-Newtonian filtration system, we will make use of the method in the second chapterto investigate (11). Because equations in (11) are doubly degenerate, we will makeuse the method of generalized to establish the existence of weak solutions of theinitial boundary value problem. In order to apply monotonicity, we establish thecomparison principle for system (11) by choosing suitable test function and Gronwall'sinequality. Then by the first eigenvalue and its corresponding eigenfunction tothe p-Laplacian operator on some domain, we construct a positive supersolution. Bythe first eigenvalue and its corresponding eigenfunctions to the eigenvalue problemfor the non-Newtonian filtration system, we construct a positive subsolution. Bychoosing suitable domains and positive constants for supersolution and subsolution,we can obtain a pair of well-ordered positive supersolution and subsolution. Usingmonotonicity methods, we obtain the existence of nontrivial nonnegative periodicsolutions. At last, we consider the set of all nonnegative periodic solutions, whichis proved that has a priori upper bound M according to the maximum norm. Bywhich we establish the existence of the maximum periodic solution and asymptoticbounds of nonnegative solutions of the initial boundary value problem.