Study On A Class Of P-Laplacian Parabolic Equations With Logarithmic Nonlinearity

In this paper,the initial boundary value problems of one type of p-Laplacian parabolic equations with logarithmic nonlinearity is considered.By using Galerkin’s method,potential well method and differential inequality techniques,the results of the existence,decay estimates and blow-up of the solution are obtained.The main purpose is to investigate the influence of the competition between the nonlinear term exponent and theΔu_tterm on the existence and blow-up of the solution.In Chapter 1,the research background and development of the related fields are intro-duced,and then the main results of this thesis is given.In Chapter 2,some notations,suitable potential well,some useful lemmas and definitions are given.In Chapter 3,the initial boundary value problem of p-Laplacian parabolic equation with logarithmic nonlinearity is studied.First,the existence of local and global solutions of the equation are proved by using the Galerkin’s method;Then combining with the potential well method,the existence of the global solution and decay estimate are obtained when the initial value energy is small enough;Finally,by constructing a suitable auxiliary functional,it is proved that the solution will grow exponentially in W₀^1,p（（?））norm as the time tends to infinity under some suitable conditions.In Chapter 4,the initial boundary value problem of p-Laplacian parabolic equation with logarithmic nonlinearity added withΔu_tterm is discussed.Firstly,we use the Galerkin’s method,Gagliardo-Nirenberg inequality and Young inequality to prove the local existence of the solution;then the global existence of the solution is obtained by using the potential well method when the initial value is in a stable set;Finally,by constructing a suitable auxiliary functional,it is studied that the solution blows up at finite time in W₀^1,p（（?））norm when the initial value satisfies suitable conditions.