The Property Of Solutions To Some Nonlinear Evolution Equations

The evolution equation is refer to the Partial differential equation depending on a continuous time variablet , which descripe the development in time of a system from physics and other science. It contains the Korteweg-de Vries (KdV) equation, Reaction-Diffusion Equations, and Equations in Fluid Mechanics as particular cases. Such PDE govern, or put in more modeste terms, model many aspects of the nature surrounding us. PDE have a significant importance for the scientific and technological progress of our society。A central issue in the study of nonlinear evolution equations is that solutionsmay exist locally in time (that is, for short times) but not globally in time. This is caused by a phenomenon called "blow-up".There are many questions which can be asked about evolution equations: the existence of particular types of solutions, such as equilibrium solutions, travelling waves, self-similar solutions, time-periodic solutions; the long time asymptotic behavior of solutions, and so on. This thesis is devoeted to the study the properties of the solutions of some nonlinear evolution equation.In Chapter 2, we considered an initial boundary value problem related to a double degenerate parabolic equation. Under suitable conditions, we establish a blow-up result for certain solution with positive initial energy. And upper bound and lower bound estimate to the blow-up time will be also considered by using the differential inequality technique.In Chapter 3, we consider the positive solution of the Cauchy problem for the above doubly degenerate parabolic equation with special source term. When the initial values decay at infinity, we give a new secondary critical exponent for the existence of global and nonglobal solutions. Furthermore, the large time behavior and the life span of solutions are also studied.In Chapter 4, we also consider the positive solution of the Cauchy problem to the above doubly degenerate parabolic equation. So many mathematicians are interested in the study to the set of blow-up points and the behaviour of solution at the blow-up point. Such problem is so difficult. In this chapter, we proved that under some conditons the blow-up set only contains the original point. At the same time, the upper and lower estimates of the blow-up solution near the single blow-up point are obtained.In Chapter 5, we study a pseudoparabolic equations which from mathematical model of nonstationary processes in semiconductors in an exterior electric field with dissipation taken into account. We derive sufficient conditions for the blow-up solutions. From the physical point of view, the blow-up of solutions corresponds to electric breakdown in semiconductors. We note that our technique of the proof of the nonexistence of global-solution is a generalization of the energy method of Levine, whose method does not directly apply to our case. Especialy, the upper bound and lower bound for the blow-up time are also determined.In Chapter 6, we consider a class of nonlocal wave equations. At first, we give the well-posed theorem. Then we give the suitable conditions such for global solution and blow-up solution.In Chapter 7, we consider how the singularitities of the potential kernel influence the existence/nonexistence of solutions of a class of aggregation equations. Our results have extended the similar results to the case of equatons with fractional diffusion term.