Qualitative Research On Nonlinear Pseudoparabolic Equations

In recent years, with the increasingly extensive application of pseudo-parabolic equations, the theory study of this issue becomes more important. The concerned problem is that the reasonability of the mathematical model in form of nonlinear pseudo-parabolic equations, which arise in many kinds of physical phenomena. For a reasonable model, the existence and uniqueness of solutions are the most important prerequisites.In this paper, the initial boundary value problem and Cauchy problem of several classes of nonlinear pseudo-parabolic equations are studied. The main contents in the paper are as follows:1. Using classical Galerkin method, the existence and uniqueness of the global strong solution for a class of pseudo-parabolic equations are proved.2. Based on the contraction mapping principle and the Fourier transformation theory, the existence and uniqueness of the global solution of two classes of nonlinear pseudo-parabolic equations are proved under certain assumptions. Then, the asymptotic behavior of the global solution is obtained.3. The asymptotic behavior of auxiliary problem which is equivalent to generalized cubic double dispersion equation is discussed by using a improved integral estimate method. Furthermore, the asymptotic behavior of the solution for the original problem is studied. Finally, applying the theorem of this paper, the asymptotic behavior of the practical problem is investigated again.