Study For A Family Of Shallow Water Equations

In this thesis,we will pay attention to a family of shallow water equations.First,we consider the Camassa-Holm equation with k≠0 on the real line.We establish certain conditions on the initial datum to guarantee the corresponding solution exists globally or blows up in finite time.Meanwhile, infinite propagation speed for is proved in the following sense:the corresponding solution u(x,t) +k with compactly supported initial datum u₀(x) +k i.e,(u₀(x) +k∈C₀^∞(R)) does not has compact x-support in its lifespan.Then,It is shown that a strong solution of the Degasperis-Procesi equation,initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time.The decay rate of the corresponding strong solution at infinity is also given for some kinds of initial data with exponential decay.We know the peakons are peaked solitary wave solutions of a certain nolinear dispersive equation that is a model in shallow water theory.We study the orbital stability problem of the peaked solitons to the special case of the DGH equation in the H¹ norm in section 4.