Blow-up Properties Of Solutions For The Nonlocal Parabolic Equation And System

In real life, a lot of phenomena in physics, chemistry and astronomy are based on mathematical models of parabolic problems with nonlocal boundary conditions, such as thermoelasticity. In this situation, solutions of the problem usually represent the material entropy per volume. This paper considers the initial boundary value problems of a nonlocal parabolic equation and a system, and studys the global existence and blow-up properties of solutions of the two problems by using the technique of super-solutions and sub-solutions.In chapter two, we first recall some basic knowledge of partial differential equation which our research needs concerning some terminologies and concepts, some frequently-used inequalities and the definition of Sobolev space. These will lay a necessary theoretical foundation for our research.In chapter three, we discuss the initial boundary value problem of nonNewtonian polytropic filtration equation with nonlocal source and weighted nonlinear nonlocal boundary condition. Because the diffusion term is nonlinear and doubly degenerate, our research is faced with new challenges. The nonNewtonian polytropic filtration includes not only the Newtonian filtration equation but also the non-Newtonian filtration equation, so the method for it should be synthetic. By investigating the impacts of the interaction between the nonlocal reaction term and the nonlinear element in the diffusion term and the different values of the weight function and the nonlinear exponent in the boundary condition on the properties of solutions, we obtain the conditions in which the solution of this problem exists globally and blows up. Furthermore, on the foundation of the deserved blow-up rate estimates, we get the blow-up profile and set of the blow-up solutions when p =2.In chapter four, we study the p-Laplacian parabolic system with nonlocal sources and weighted linear nonlocal boundary conditions. First, we define the super-solution and sub-solution of this problem, and establish a modified comparison principle. Then by the methods of the elliptic equations and ordinary differential equations, we construct proper super- solutions and sub-solutions and observe the influence of interaction among the boundary condition, the nonlinear diffusion term and the nonlocal reaction term on blowing up or not of solutions. Moreover, we obtain when the solutions of this problem exist globally and the conditions in which the solutions blow up.