Dynamical Properties Of Solutions To A Class Of Weakly Dissipative Nonlinear Shallow Water Wave Equations

In this thesis, a nonlinear shallow water wave equation with weaklydissipative term, which includes weakly dissipative Camassa-Holm and weaklydissipative Degasperis-Procesi equations as special cases, is investigated. Undersuitable assumptions, some dynamical properties for the equation, which includenot only local existence and uniqueness of solutions, global strong solutions,blow-up solutions and well-posedness of global weak in the sense of distributionsbut also persistence properties, unique continuation and infinite propagationspeed of strong solutions and so on, are acquired.The main contents of the work are organized as follows.Chapter 1 focuses on the global solutions and the blow-up phenomena to theequation (0-1), the local well-posedness of solutions to the equation with initialvalue u0∈Hs (s > 23) is established by using the Kato theorem. It is shownthat provided that initial value u0∈Hs L1(R) and y0 = (1-( ?)_x²)u₀ changessign, the blow-up phenomena occur in finite time. Moreover, a global existenceof strong solutions of the equation for certain initial profiles is obtained.In chapter 2, we discuss some properties of solutions to the equation (0-1),which are not studied in the chapter 1. Provided that u0∈Hs L1(R) andy0 = (1-(?)x₂)u0 is of one sign, the global existence of strong solutions is acquired and the global solutions decay to 0 in the Hs-norm space as time goes to infinity.The persistence properties, unique continuation and infinite propagation speedof strong solutions are also investigated under suitable assumptions.Chapter 3 concerns the existence and uniqueness of global weak solutionin the sense of distributions for the equation (0-1). Assume that (1 - (?)x2)u0∈M+(R), u0∈H1(R) L1(R), the well-posedness of global weak solutions forthe equation is shown to be true.In chapter 4, the global weak solutions for the Degasperis-Procesiequation with weakly dissipative term are investigated. Provided that(1 -(?)_x²)u₀∈M+(R), u0∈H1(R), suppy0(?) (-∞,x0) suppy0+ (x₀,∞), theexistence and uniqueness of global weak solutions in the sense of distributionsfor the equation are proved to be valid. It should be addressed that theassumptions u0∈L1(R) used in chapter 3 is cancelled in this chapter.