Interactions Between Localized-local Sources And Asymptotic Behavior Of Singular Solutions

This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equation (system) with local and localized sources. Five models with local and localized sources of the sum forms are involved. We will study interactions between the two kinds of sources and their influences to the occurrence and propagation of singularities of solutions, and pay more attentions to the topic of blow-up sets. At first, we consider global versus single point blow-up of solutions to two models with coupling localized and local sources respectively. Comparing the two models, we find substantial different influences of the two kinds of couplings to the blow-up sets of solutions. Then, we study in what way the interactions between localized and local sources affect the blow-up rate, blow-up profile, and blow-up set in a nonlinear diffusion problem. Finally, we will show via models how the localized sources substantially influence the critical Fujita exponents. It is interesting to find they may admit an infinite Fujita exponent because of the localized sources. This excludes the situation where the solutions are non-global for large initial data and global with small initial data.The main results obtained in this thesis can be summarized as follows:(â… ) Total and single point blow-upChapter 2 considers heat equations with local and coupling localized sources u_t =â–³u + u^m + v^p(0, t), v_t =â–³v + vⁿ + u^q(0, t), (x, t)∈Ω×(0, T) subject to null Dirichlet boundary conditions. The behavior of solutions depends on the interactions among the local and localized sources as well as the diffusions with the null boundary conditions in the model. We obtain a complete classification of parameters to distinguish total and single point blow-up for the non-global solutions. In addition, simultaneous versus nonsimultaneous blow-up of solutions under different dominations are determined also with four possible simultaneous blow-up rates. To our knowledge, this is the first study on total versus single point blow-up for the case of coupled systems.Chapter 3 treats homogeneous Dirichlet problem to heat system with localized and coupling local sources u_t =â–³u + u^m(0,t) + v^p, v_t =â–³v + vⁿ(0,t)+ u^q, (x, t)∈Ω×(0, T) with a parallel discussion as that in Chapter 2, i.e., total versus single point blow-up, simultaneous versus non-simultaneous blow-up etc. In particular, comparing with the results of Chapter 2, we find some substantial differences on occurrence and propagation of singularities of solutions due to the two kinds of couplings. For example, the situation of global and single blow-up for the two components respectively, included in the classification to the model in Chapter 2, does not appear in the classification to the present model.Chapter 4 studies a localized nonlinear diffusion equation u_t =â–³u^m+λ₁u^p+λ₂u^q(0, t) subject to null Dirichlet boundary condition with p, q≥0, max{p, q} > m > 1, andλ₁,λ₂ > 0. By investigating the interactions among the localized and local sources, the nonlinear diffusion with the zero boundary value condition, we establish blow-up rates and uniform blow-up profiles of solutions under different dominations. In addition, as for the blow-up sets of solutions, we find that nonlinear diffusion has no contributions to the total and single point blow-up of solutions.(â…¡) Fujita exponentsChapter 5 deals with Cauchy problem to nonlinear diffusion model u_t =â–³u^m +λ₁u^p₁(x,t) +λ₂u^p₂(x^*(t),t) with m≥1, p_i,λ_i≥0 (i = 1,2) and x^*(t) H(?)lder continuous. A new phenomenon is observed that the critical Fujita exponent p_c= +∞wheneverλ₂ > 0. More precisely, the solution blows up under any nontrivial and nonnegative initial data whenever p = max{p₁,p₂} > 1. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities (even the decay ones).Chapter 6 focuses on the asymptotic behavior of solutions for reaction-diffusion equations coupled via localized and local sources: u_t =â–³u+v^p(x^*(t),t). v_t =â–³v+u^q Both the initial-boundary problem with null Dirichlet boundary condition and the Cauchy problem are considered to study the interaction between the localized and the local sources. For the initial-boundary problem we prove that the solutions blow up everywhere in the domain with uniform blow-up profiles. In addition, it is interesting to show that the Cauchy problem admits an infinity Fujita exponent, namely, the solutions blow up under any nontrivial and nonnegative initial data whenever pq > 1.