Oscillation Of Partial Functional Equations And Blow-up For Nonlinear Reaction-diffusion Equations

With Development of modern science and technology, there are many time delayed phenomena in mechanics, biology heredity engineering, cybernetics and populations dynamics. And the models of mathematics simulation of such processes and phenomena are partial differential equations with functional arguments, which are called partial functional differential equations. The oscillation theory is the one of the important branches of qualitative theory of partial functional differential equations. Therefore, it is of great theoretical and practical value to research the oscillation of partial functional differential equations.Nonlinear reaction-diffusion equations come from many mathematical models in physics, chemistry and biology, which have strongly practical background.In this paper, we will consider the oscillation of partial functional differential equations and impulsive partial functional differential equations, and the global existence and blow-up for the nonlinear reaction-diffusion system.In chapter 1, we will introduce the oscillation theory of partial functional differential equations and the basic concepts of blow-up for nonlinear reaction-diffusion, and summarize the recently important research results about the two aspects, including my research work.In chapter 2, we discuss the oscillation of neutral partial functional differential equation (system). We introduce the results in basic theory of partial differential equations for boundary eigenvalue problem that the smallest eigenvalue and the corresponding eigenfunctions are positive, and obtain the sufficient conditions for the oscillatory solutions under the corresponding boundary conditions. Furthermore, we discuss the oscillationof a class of neutral partial function differential system. By some techniques in differential inequalities, we consider the oscillation of a class of hyperbolic system with continuous distributed arguments under the corresponding boundary conditions and the sufficient conditions are obtained.In chapter 3, we investigate the oscillation for impulsive partial functional differential equations. We transform the oscillation problem to the one that the impulsive delayed ordinary differential inequalities don't exist the eventual positive solutions. By impulsive differential inequalities, we investigate oscillation of impulsive multirdelayed hyperbolic equations under the corresponding boundary conditions, and some oscillatory criteria are obtained.In chapter 4, we consider global existence and blow-up for nonlinear reaction-diffusion equations. We consider a class of reaction-diffusion system with nonlinear memory under the homogenous Dirichlet boundary conditions, and obtain the conditions of global existence and blow-up of the solutions. Firstly, we establish the corresponding comparison principle, and use the properties of the eigenfunction and the self-similar solutions to construct the global super-solutions and the blow-up sub-solutions. By the sup-sub solutions technique, we obtain the sufficient conditions of global existence or blow-up in finite time. Furthermore, under some appropriate hypotheses, we obtain the blow-up rate estimates of blow-up solutions.In chapter 5, we discuss the blow-up for nonlinear degenerate reaction-diffusion system. Firstly, we establish the local existence and uniqueness of the weak solutions. Then, we construct the appropriate sup-solutions and sub-solutions by the comparison principle, and obtain the conditions of global existence and blow-up in finite time. Finally, we estimate the blow-up rate of the solutions.