| In this thesis we study several properties of solutions to Cauchy problems for a class of shallow water wave equations, including local well-posedness, blow-up, global existence of strong solutions and weak solutions and so on. These equations arise from modern mechanics and physics, and they are important subjects in the study of nonlinear science, especially in the field of soliton and integrable system. This thesis is organized as follows.In Chapter 1, we outline the background and research status.In Chapter 2, we consider the Cauchy problem for a weakly dissipative generalized μ-Hunter-Saxton equation, and establish local well-posedness, blow-up criterion and blow-up rate of strong solutions, Holder continuity of the solution map, and global existence of strong solu-tions and weak solutions. Our main aim is to study the effects of free parameter σ and dissipative parameter λ in blow-up and global existence results. Applying the Littlewood-Paley theory or Kato’s semigroup theory, we first show that the equation is locally well-posed for any initial data u0∈Hs(S), s>3/2. Then the blow-up criterion and blow-up rate are discussed in terms of the sign of σ by energy method. We give an explicit lower bound for the maximal existence time T with respect to ‖u0‖Hs(S) and an estimate of the solution size, and then we prove that the solution map is Holder continuous in Hs(S), s≥2, equipped with the Hr(S)-topology for 0≤r<s. Moreover, global existence of strong solutions is discussed according to the sign of A/σ. Finally, using the viscous approximate method, we establish the existence of global weak solutions in H1(S). Different from the previous works, an Oleinik-type estimate is not needed here. Since this estimate is not easy to be verified in numerical experiment, the improvement is meaningful.In Chapter 3, we discuss the Cauchy problem for a weakly dissipative generalized two-component μ-Hunter-Saxton system, and present local well-posedness, blow-up criterion and blow-up rate of strong solutions, and global existence of strong solutions and weak solutions. The purpose is to investigate the effects of free parameter σ and dissipative parameter A in blow-up and global existence results for the two-component case. Similar as in the Chapter 2, we first obtain the local well-posedness in Hs(S)×Hs-1(S), s>3/2. Then the blow-up criterion and blow-up rate are established in terms of the sign of σ, and a lower bound of the lifespan for σ>0 is provided. We finally show that the strong solutions exist globally in time for 0≤σ< 2, by constructing the corresponding Lyapunov functions.Chapter 4 is devoted to studying the Cauchy problem for a modified two-component Camassa-Holm system in higher dimensions. Using the Littlewood-Paley analysis and the trans- port equations theory, we first establish the local well-posedness in both supercritical and critical nonhomogeneous Besov spaces, which imply the local well-posedness in Sobolev spaces. We then present a blow-up criterion of strong solutions by energy method. Moreover, zero density limit and zero dispersion limit for weak solutions are investigated, and finally a Liouville type theorem for the stationary weak solution is provided. |
