Study On Solution Properties Of A Class Of Pseudo-parabolic Equations With Logarithmic Terms

Pseudo-parabolic equations are often used to simulate various natural phenomena and physical processes,such as characterization of the unidirectional propagation of nonlinear,dispersive and long waves,description of population migration,research on semiconductor technology in physics,two phase flow in porous media with dynamical capillary pressure and so on.Therefore,the corresponding practical problems can be analyzed qualitatively and quantitatively by using mathematical theory and method to study it systematically.This thesis deals with the initial boundary value problem for a pseudo-parabolic equation with p-Laplacian and logarithmic nonlinearity terms,and we consider some properties about weak solution,such as global existence,asymptotic behavior,blow-up in finite time,a lower and an upper bound estimation for blow-up time and rate and so on.This thesis mainly contains the following four sections:In Section 1,this thesis mainly introduces the development background of a pseudo-parabolic equation with logarithmic nonlinearity terms,and introduces the research purpose,research method and definition of some symbols.In Section 2,studying asymptotic behavior of the global bounded solution corresponding initial boundary value problem,we prove that the weak solution converges to the corresponding stationary solution as time tends to infinity under the proper condition.In Section 3,the initial conditions that we constructed make the solution of the corresponding initial boundary value problem exist globally or blow up.Firstly,we give conditions such that the weak solution blow up in different initial energy cases,and estimate blow-up time and rate.Next,sufficient condition is given to make the weak solution globally bounded,and it is proved that the weak solution converges to zero when time tends to infinity.Finally,We give an initial condition which is independent of the energy,and obtain nonblow-up and algebraic decay in H ₀¹(?)-norm under this condition.In Section 4,Summarizing the main content and innovation of this thesis,we put forward some questions to be solved,and look forward to the future research work.