The Analysis Of The Solutions To Several Diffusion Equations (Systems)

A variety diffusion phenomena appear wildly in nature. In applied science, manyproblems related to diffusion can be modelled by partial differential equations with dif-fusion terms. For instance, these equations arises in many fields such as filtration, phasetransition, biochemistry and dynamics of biological groups. In views of mathematicaltheory and applied sciences, it is very important to analysis these diffusion equations,which brings out a series new mathematical ideas and methods. In the last four decades,especially in recent twenty years or so, the study in this direction attracts a large numberof mathematician both in China and abroad. Remarkable progress has been achieved.Many new ideas and methods have been developed, which enrich enormously the theoryof partial differential equations. In this thesis, we will give some qualitative analysis forseveral diffusion equations (systems) arose in applied sciences.In Chapter 1, we will investigate several reaction-diffusion systems with nonlo-cal/localized sources subject to Neumann boundary conditions and Cauchy initial data.For the Neumann boundary value problems, we use some new comparison principles toobtain the sufficient conditions that the solutions exist globally or blow up in a finite time.At the aid of Green function, we also establish the precise blowup rate estimates and getthe blowup set. We will see the blowup estimates hold uniformly in the whole domainbecause of the isolated effect of homogeneous Neumann boundary. Using similar ideasand the technique of ODE, we will also consider the Cauchy problem for the correspond-ing systems and study the simultaneous blow-up and non-simultaneous blow-up, blowupestimates and so on. These results improve and extend the previous related works.In Chapter 2, we first deal with the homogenous Dirichlet boundary-value problemfor a diffusion system with reaction and absorption. Under the suitable growth restric-...