The Study Of Well-posedness And Bifurcation Phenomena Of Several Kinds Of Nonlinear Equations

In this paper, we mainly study well-posedness and bifurcation phenomena of several nonlinear equations. On the one hand, by some basic facts on the Littlewood-Paley theory and the transport equations theory, we proved the local well-posedness of quasi-linear evolution equation and hyperelastic rod equation in Besov space Bp,rs. Furthermore, we study the continuity of the solution map. On the other hand, by exploiting the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtained many new traveling wave solutions and bifurcation phenomena for Schamel-Korteweg-de Vries equation and Zakharov-Kuznetsov equation. The main work of this dissertation are as follows.In Chapter 1, we mainly introduce the background, research developments, achieve-ments of our research objects, and the main content of this paper.In Chapter 2, we study the periodic boundary value problem for a quasi-linear evolution equation of the following type: (?)tu+f(u)(?)xu+F(u)= 0, x ∈T= R/2πZ, t E R+ Under some conditions, we prove that this problem is locally well-posed in Besov space Bp,rs(T). Furthermore, we study the continuity of the solution map in B2,rs(T).In Chapter 3, we study the periodic boundary value problem for hyperelastic rod equation of the following type: (?)tu-(?)t(?)x2u+3u(?)xu=γ(2(?)xu(?)x2u+u(?)x3u), x∈T, t∈R+. We prove this problem is locally well-posed in the critical Besov space B,1 3/2 or in Bp,r s with 1≤p, r≤+∞, s> max{1+1/p,3/2}. We also prove that if a weaker Bp,rq-topology is used, then the solution map becomes Holder continuous. Furthermore, we show that the solution map is not uniformly continuous in B2,rs with 1≤p, r≤+∞, s> 3/2 or r=1,s=3/2.In Chapter 4, we study the nonlinear waves described by Schamel-Korteweg-de Vries equation. Two new types of nonlinear waves called compacton-like waves and kink-like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell-shape solitary waves, the blow-up waves, the valley-shape solitary waves, the kink-like waves and the compacton-like waves.The second phenomenon is that the periodic-blow-up wave can be bifurcated from the smooth periodic wave.In Chapter 5, we study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equation. We reveal four kinds of bifurcation phenom-ena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves and the anti-symmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.