Asymptotic Behavior Of Solutions To Degenerate Parabolic Equations Not In Divergence Form

This thesis deals with asymptotic behavior of solutions to degenerate parabolic equa-tions not in divergence form. The topics include critical exponent and large time behavior for two parabolic equations with source and absorption not in divergence form, the influ-ences of the gradient term on the solutions for a doubly degenerate parabolic equation, as well as the blow-up conditions for a strong coupled degenerate parabolic system. All the four models involved are multi-nonlinear with nonlinear diffusion not in divergence form. At first, we consider two models with source and absorption, where a precise analysis on interactions among the multi-nonlinearities is given to determine the critical exponent and the large time behavior of solutions. Then, we study in what way the additional gradient term affects the behavior of solutions in a doubly degenerate diffusion equation. Finally, we establish the blow-up conditions and the blow-up set for a strongly coupled degenerate parabolic system with localized source.The thesis is composed of four chapters:Chapter1summarizes the background of the related issues, and briefly introduces the main results of the present thesis.In Chapter2, we at first consider the problem ut=upΔu+aug—bur with homoge-neous Dirichlet boundary. We obtain the critical exponent by using the extended Kaplan method, and then study the large time behavior for the global solutions. When the posi-tive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption. Secondly, we discuss the problem ut=upΔu+uq|▽u|2—ur to determine the uniform decay rate of solutions.Chapter3study a doubly degenerate equation of the form ut=umdiv(|▽u|p-2▽u)+λuq+γur|▽u|p with the null Dirichlet boundary condition. We at first establish the local existence of weak solutions to the problem, and then determine in what way the gradient term affects the behavior of solutions. The conditions for global and non-global solutions are obtained with the critical exponent rc=pm-q/p-1. In Chapter4,we investigate the blow-up condition and blow-set for the strong cou-pled degenerate parabolic system ut=υp(△u+af(u(xo,t))),uq=(△υ+bg(υ(xo,t))) with null Dirichlet boundary.