Nonlinear Parabolic Equations With Weighted Nonlocal Sources

This thesis mainly deals with nonlinear parabolic equations with weighted nonlocal sources. The topics include influences of weight functions in nonlocal boundary conditions to the blow-up behavior of solutions for coupled nonlocal parabolic equations, effects of weight functions in nonlocal sources to blow-up sets and blow-up rates of solutions, and in addition, influences of asymmetric sources and inner absorptions to the asymptotic behavior of solutions, and so on. Firstly, we consider a parabolic system coupled via nonlocal sources where each component on the boundary takes its weighted mean value over the domain. An exponent classification for blow-up and global existence of solutions is obtained, where one can find the substantial contribution of the weight functionsφandψto the blow-up behavior of solutions. Secondly, we concern a nonlinear parabolic equation with a weighted localized source, where the blow-up rate estimates obtained do depend on the weight function. Thirdly, we study a parabolic system with asymmetric nonlinearities to observe the influence of the asymmetric sources to blow-up profiles of solutions. In addition, the thesis deals with the asymptotic analysis to a parabolic models with inner absorptions and weighted localized sources. Uniform blow-up profiles are established for the case of weak absorptions. While for the two cases of unbalanced absorptions, instead of blow-up profiles, the blow-up rate estimates are established, which are absorption-related, and hence substantially different from scalar parabolic problems with inner absorptions in the current literatures (where all the blow-up rates are known as absorption-independent).Chapter 1 is an introduction to recall the background of the related topics and summarize the main results of the present thesis.Chapter 2 deals with a semi-linear parabolic system with coupled nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. We establish the conditions for global and non-global solutions respectively. It is interesting to observe that the weight functions in the nonlocal Dirichlet boundary conditions have a substantial contribution to determine not only the solutions are global or non-global, and but also (for the non-global solutions) the blowing up occurs for any positive initial data or just for large ones. Chapter 3 studies a nonlinear parabolic equation with a more complicated source term, which is a product of localized source u~q(0,t), local source u~p(x,t), and weight function a(x). We investigate how the three factors influence the asymptotic behavior of solutions. We at first determine a complete classification for single point versus total blowup of solutions, and then establish some a(x)-related lower bound estimates of blow-up rates for total blow-up case.Chapter 4 concerns a nonlinear parabolic system subject to null Dirichlet boundary conditions, where the coupled nonlocal sources consist of mixed type asymmetric nonlinearities. We at first obtain the sufficient and the necessary conditions respectively for simultaneous blow-up in the model, and then establish uniform blow-up profiles of solutions near the blow-up time. It is interesting to observe that not only the simultaneous blow-up rates of the components u and v are asymmetric, but also the blow-up rates of the same component u (or v) may be in different levels under different dominations.Chapter 5 focuses on the asymptotic analysis to a parabolic model with inner absorptions and weighted localized sources. Three possible simultaneous blow-up rates under different dominations of nonlinearities are established. In particular, uniform blow-up profiles are established for the case of weak absorptions. While for the two cases of unbalanced absorptions, instead of blow-up profiles, the blow-up rate estimates are established, which are absorption-related, substantially different from those for all the scalar problems with inner absorptions .