Study On Dynamical Behavior To One Shallow Water Wave Equation

In this doctoral dissertation, we study the local well-posedness, precise blow-up scenario and blow-up phenomena, global existence of strong solutions to the Cauchy problem for a class of nonlinear shallow water wave equations and the global attractor, numerical simulation of MKdV equation.The dissertation is divided into six chapters. In the first chapter, we first introduce the physical background and the latest research advances of the equations. We then give some related definitions and notations of the dissertation. In the second chapter, we study the Cauchy problem of a generalized dissipative CH equation. We first establish the local well-posedness for the equation on Hs(R),s>3/2by Kato’s theory. Then we present a precise blow-up scenario of strong solutions to the equation, and give several blow-up results by using some priori estimates and several useful lemmas. Finally, we prove that the blow-up rate of strong solutions to the equation is-2. This fact shows that the blow-up rate is not affected by the weakly dissipative term, but the occurrence of blow-up is affected by the dissipative parameter A. In the third chapter, we study a weakly dissipative two-component CH equation. We investigate the local well-posedness of the Cauchy problem for any initial data z0=(u0,ρ0)∈Hs(R)×Hs-1(R), s≥2by Kato’s theory. Then we also present a precise blow-up scenario, several blow-up results and the blow-up rate of strong solutions to the equation. In fourth chapter, we study the Cauchy problem of a dissipative two-component CH equation with free parameter σ. We obtain the local well-posedness, precise blow-up scenario, blow-up phenomena and the blow-up rate of the equation. In fifth chapter, we study a weakly dissipative periodic two-component CH equation. We first obtain the local well-posedness, precise blow-up scenario, blow-up phenomena and the blow-up rate of the equation. Then we obtain a sufficient condition for the existence of global solution of the equation when0<σ<2by using Lyapunov function. In last chapter, the Sovolev interpolation inequality and prior estimate on time t are applied to show the existence of solution in unbounded domain. Moreover, operator decomposed technique and Kuratowskii α-noncompacted measure are applied to study the smooth property of the solution, then the existence of the global attractor of a dissipative MKdV equation in H2(R) is proved. Finally, we use nature analysis approach and dissipative conservative scheme theory to prove the computational stability of universal double time differential format and the accordant conclusion with theoretical proof is gained by numerical simulation.