On The Limit Behavior And Well-Posedness Problem For A New Class Of Shallow Water Wave Equations With Dispersive Term

In this paper, we study the limit bahavior and the well-posedness of Cauchy problem for a new nonlinear dispersive shallow water wave equations with dispersive term. DGH (the so-called Dullin-Gottwald-Holm equation, i. e., DGH equation) equation is the 1+1 quadratically nonlinear equation for unidirectional water waves, which was derived by Dullin, Gottwald and Holm, by using asymptotic expansions directly in the Hamiltonian for Euler's equations in the irrotational incompressible flow of a shallow layer of inviscid fluid moving under the fluence of gravity as well as surface tension. Degasperis and Procesi studied the D-P(Degasperis-Procesi, i. e., DP equation) equation. They found that there are only three equations that satisfy the asymptotic integradility condition with this family: the KdV equation, the Camassa-Holme equation, and the Degasperis-Procesi equation.Chapter two studies the local well-posedness, blow-up theory of the initial value problem of DGH equation with dispersive term. Rewriting the nonlocal form of the equation, using the Kato' s theory, we obtain the local well-posedness of the equation; After the discussion about singularity of the initial value problem, a sufficient condition of blow-up is obtained. Chapter three studies the integrability of a class of nonlinear dispersive shallow water wave equation with dispersive term (DGH equation and D-P equation with dispersive term). By using the Symmetry Approach and Number Theory methods, the evolution partial differential equation is studied. The integrability of nonlinear dispersive shallow water wave equation can get from the integrability of the homogeneous differential polynomial. Chapter four shows the relation between the solution of initial value problem of DGH equation with dispersive term and that of Camassa-Holm equation. Under the discussion about general solution of DGH equation with dispersive term, we prove that when dispersive parameterγconverges to zero, the solution of DGH equation converges to that of Camassa-Holm equation. Chapter five studies the local well-posedness and the global well-posedn ess of the initial value problem of D-P equation with dispersive term. Rewriting the nonlo-cal form the equation, using the Kato' s theory, we obtain the local well-posedn ess of D-P equation with dispersive term; according to some prior estimates which prove that the solution has global well-posedness when positive property is given for initial potential.