Study On Two Kinds Of Nonlinear Parabolic Equations

In this thesis,the potential well theory,Galerkin method and convex method are used to study the initial boundary value problem for two classes of nonlinear parabolic equations with a memory term and nonlinear source term.It is devoted to reveal the influence of the initial conditions,exponential relation and source term on the properties of the solutions.In the first chapter,the research background and significance of parabolic equation,as well as the research status at home and abroad are introduced.In Chapter 2,the well posedness of solutions for a class of nonlinear parabolic equations with memory term is discussed.The existence of local solution is proved by means of the Galerkin method and Sobolev embedding theorem.On the basis of its solution,considering the influence of the exponential relation on the properties of the solution,combining with the initial energy and auxiliary function,it is established that the solution grows exponentially in the L~pnorm,and it is shown that the global existence of the solution exists by using the continuity principle and the modified energy functional.Finally,with the help of potential well theory and considering the suitable initial conditions,the existence of the global solution is attained,and the energy decay estimate of the solution is also given.In Chapter 3,the initial boundary value problem for a class of pseudo-parabolic equations with logarithmic nonlinear source term and memory term is considered.By using logarithmic Sobolev inequality and potential well theory,the global existence and asymptotic behaviour of solutions with respect to the influence of the initial data is acquired.Combined with the convex method,the blow up and blow up rate at infinite time are obtained.