Some Properties Of Solutions For Several Parabolic Equations With Weighed Non-local And Local Sources

This thesis mainly studies some properties of three class parabolic equations with homogeneous Dirichlet boundary conditions, including existences, blow-up and blow-up rates. This thesis consists of five chapters:the first two chapters are introduction and some basic knowledge. In the introduction, the background and current research for the three parabolic problems are given. In the second chapter, some basic knowledge to be used in this thesis is given. The following three parabolic systems are discussed in the next three chapters separately:In the third chapter, the global existence and blow-up properties of solutions for parabolic system (3.1) with weighted non-local sources and absorptions are considered. By using the super-solution and sub-solution method, the sufficient conditions of global existence and blow up in finite time of solutions are obtained, and then the global blow-up rates related to a(x),b(x)are proved.In the forth chapter, the existence of weak solution of the nonlinear parabolic system (4.1) with double variable exponents is discussed. By using the methods of parabolic regularization and Galerkin’s approximations, the sufficient conditions which guarantee the existence of bounded weak solutions are established in suitable Sobolev-Orlicz spaces.In the fifth chapter, the blow-up properties of solutions for parabolic system (5.1) coupled via different sources with nonstandard growth conditions is investigated. Firstly, the local existence of solutions is proved by modifying the boundary conditions; then, a sufficient condition of global existence of solutions is obtained; forΩ= BR(0), not only the complete classification on solutions in simultaneous and non-simultaneous blow-up is obtained, but also the blow-up rates are obtained.