Asymptotic Behaviors Of Solutions To Reaction-Convection-Diffusion Equations And Systems

This thesis deals with the large time behavior of solutions to nonlineardifusion equations and systems with reactions and convections. Further-more, the coefcients of reactions and convections may be singular anddegenerate. The thesis contains two parts.In the frst part, we study the non-Newtonian polytropic fltrationequations with reactions and convections. As a typical kind of quasilineardifusion equations, these equations have been studied extensively. Sincethe convections of the considered equations are singular at the origin, westudy the homogeneous Neumann exterior problems. We obtain the Fu-jita exponents and establish the theorems of Fujita type for the problems. The results display how the degenerate nonlinear difusions, the nonlinearconvections with degenerate coefcients and the nonlinear reactions withsingular coefcients influence the large time behavior of solutions to theequations. Particularly, it is shown that the Fujita exponent can be theinfnite due to the influence of convections. There are self-similar superand sub solutions for most nonlinear difusion equations. Many theoremsof Fujita type are proved by the method of super and sub solutions, whilethe critical case is treated by the method of the asymptotic profle analysis.For the considered equations in this part of the thesis, the coefcients of thenonlinear convections and the nonlinear reactions destroy the self-similarstructure and some estimates are needed when constructing self-similarsubsolutions. Moreover, one can get suitable self-similar subsolutions forthe blowing-up theorem only in some cases. So, we use the method ofthe asymptotic profle analysis to prove the blowing-up theorem. Due tothe existence of the convections and the reactions, the asymptotic pro-fle of solutions is much complicated. By precise estimates, we prove theblowing-up theorem. The existence of nontrivial global solutions is shownby constructing self-similar supersolutions.In the second part, we study a class of semilinear reaction-convection-difusion systems with singular and degenerate coefcients, which are cou- pled by the reactions. Such systems can be used to describe some difusionphenomena coming from physics and biology. Since the convections of theconsidered systems are singular at the origin, we study the homogeneousNeumann and Dirichlet exterior problems. The critical Fujita curves aredetermined and the theorems of Fujita type are established. The resultsdisplay how the difusions, the convections with degenerate coefcients andthe nonlinear reactions with singular coefcients influence the large timebehavior of solutions to the systems. In particular, it is shown that thecritical Fujita curve can be the infnite due to the influence of convections.We use the method of the integral estimate to prove the blowing-up theo-rem instead of the method of subsolutions. The merit of this method lies inthat it is not a pointwise comparison but an energy comparison. The key ishow to construct suitable weights for the energy comparison. For a systemcoupled by reactions, solutions to an equation will influence solutions toanother equation. So, the weights for energy estimates should possess thesame support. In this part of the thesis, we should choose diferent weightsfor diferent convections and rescale the weights according to the convec-tions and reactions so that the energy integrals for the two solutions areof the same grade. By constructing the suitable weights and doing preciseestimates, we prove the blowing-up theorem. The existence of nontrivial global solutions is shown by constructing self-similar supersolutions.We establish the theorems of Fujita type for some nonlinear difu-sion equations and systems with reactions and convections, which displayhow the difusions, the convections and the reactions influence the largetime behavior of solutions to nonlinear difusion equations and systems.The methods and ideas in this thesis can be used to study the large timebehavior of solutions to other nonlinear difusion equations and systems,especially the ones with convections.