Analysis On Singular Solutions To Quasilinear Parabolic Equations

This thesis deals with singularity properties of solutions, such as blow-up, extinction and quenching, to quasilinear parabolic equations. The topics include Fujita type theo-rems for fast diffusion evolution equations with variable exponents, blow-up and extinction conditions for fast diffusion p-Laplace equations with Neumann condition, simultaneous and non-simultaneous quenching in coupled quasilinear parabolic systems. The models may involve multi-nonlinearities such as nonlinear diffusion, nonlinear reaction, nonlinear, boundary flux. The blow-up criteria and asymptotic behaviors of solutions are determined by the interactions among all these nonlinearities.The thesis is composed of four chapters:In Chapter 1, we summarize the background of the related issues and state the outline of the present thesis.In Chapter 2, we study the critical Fujita exponent for the Cauchy problem of fast diffusion equation with variable exponent ut=△um+up(x). In particular, we obtain some Fujita type conditions as well for the Dirichlet problem in a bounded domain.In Chapter 3, we consider blow-up and extinction conditions for the Neumann prob-lem to fast diffusion p-Laplace equations with local source ut-div(|▽u|p-2▽u)=|u|q-FΩ|u|qdx, and the blow-up condition for the Neumann problem to slow diffusion p-Laplace equation with nonlocal source ut-ut-div(|▽u|p-2▽u)=|u|q-1u-fΩ|u|q-1udx.In Chapter 4, we investigate the blow-up condition and blow-up rate of solutions for coupled porous medium system ut=△um+aua∫Ωupdx,ut=△um+buβ∫Ωuqdx with weighted nonlocal boundary conditions, as well as the simultaneous and non-simultaneous quenching criterion for the porous medium system coupled via nonlinear boundary ab-sorptions.