Qualitative Properties Of Solutions For Parabolic Equations With Nonlocal Terms

In this paper, we study the qualitative properties of solutions for three parabolic equations with nonlocal terms. The main results include global existence, decay estimate and finite time blow-up of the solutions.Chapter 1. Investigate a slow diffusion equation with nonlocal source and inner absorption subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. Based on an auxiliary function method and a differential inequality technique, lower bounds for the blow-up time are given if the blow-up occurs in finite time.Chapter 2. Consider initial-boundary value problem of a fourth-order parabolic equation with finite memory and a generalized Lewis function which depends on both spacial variable and time subject to Neumann boundary condition. Firstly, by combining the Faedo-Galerkin method, contraction mapping principle and contradiction, obtain the local existence and uniqueness of weak solution. Secondly, by constructing a stable set, we gain the global existence of weak solution and energy uniform decay estimates with positive initial energy. Finally, construct an unstable set and conclude blow-up of solution with small positive or non-positive initial energy.Chapter 3. Deal with initial-boundary value problem of a semilinear heat equation with past and finite history memories and a generalized Lewis function which depends on both spacial variable and time subject to null Dirichlet boundary condition. Improve research methods used in chapter 2 (the existence of past memory brings difficulty), deduce global existence and uniqueness and a general decay estimate of weak solution.