Invariant Set And Exact Solutions To The (3+1)-dimensional Wave Equations And The High-dimensional Quasilinear Heat Equations

With the development of linear theory, nonlinear science is applied in the field of physics, chemistry, biology and information science. Nonlinear partial differential equation is a kind of mathematical physics model, which solves the problem arised in various fields. It is of great importance to analyze its exact solutions theoretically and practically.In this dissertation, we study the (3+1)-dimensional wave equation and high-dimensional quasilinear heat equation by utilizing the method of invariant set. The structure of this is organized as follows:In chapter 1, we briefly describe the main contents of the research back-ground of nonlinear partial differential equations, methods and background of this article.In chapter 2, we introduce the function invariant set and study the (3+1)- dimensional wave equation utt= A(u)uxx+B(u)uyy+C(u)uzz+D(u)ux2+G(u)uy2+P(u)uz2+Q(u), where A(u), B(u), C(u), D(u), G(u), P(u), Q(u) are all sufficiently smooth functions, then some new exact solutions of this wave equation are obtained.In chapter 3, we firstly establish the function invariant set and then we study the (3+1)- dimensional Quasilinear heat equation where A(u), B(u), C(u), D(u), G(u), P(u), Q(u) all sufficiently smooth functions, then some new exact solutions of this equation are obtained. Sec-ondly, we briefly discuss the (N+1)- dimensional quasilinear heat equation, and give some exact solutions.In chapter 4, we summarize our work and give some future work.