On Properties Of The Solutions Of A Class Of Nonlinear Shallow Water Wave Equations

The nonlinear evolution equations come from physics,chemistry,biology and other disciplines to build models and solve practical problems,so these equations have deep background and wide application.Nonlinear shallow water wave equations are an important kind of nonlinear evolution equations.In recent years,such equations have been one of the most popular research branches in mathematics.In this thesis,we mainly consider the properties of the solutions to the initial value problems associated with the high-order Camassa-Holm equation,the two-coupled Camassa-Holm system,the two-component Novikov system,the Geng-Xue system,and the mutli-dimensional Camassa-Holm-type system.In order to make the relationships between the initial value and the corresponding solutions clearer,it is in the periodic or non-periodic case that we obtain the non-uniformly continuous dependence on initial data of the solutions to these nonlinear water wave equations in Besov spaces,which is based on the local well-posedness results.In other words,the solution map of these problems is not uniformly continuous in the corresponding energy spaces.We mainly give the proof of the results by means of the method of approximate solutions.At first,the appropriate approximate solutions are constructed.Next,we assume that the initial values are equal to the approximate solutions when t=0,so that the relationship between approximate solutions and solutions can be set up with the initial value as the bridge.Then,the difference between approximate solutions and actual solutions is estimated.In the end,we obtain the corresponding results by the interpolation inequality and related theories in energy spaces.Thus,the constructions of the approximate solutions seem crucial in our proof.Besides,on the basis of the local well-posedness results,the Holder continuity of the solution map to the high-order Camassa-Holm equation,the two-coupled Camassa-Holm system and the Geng-Xue system in the corresponding energy spaces are discussed in detail.In addition to the above properties,the local regularity and analyticity in Sobolev-Gevrey spaces to the problem associated with the two-component Novikov system are investigated by making use of the generalized Ovsyannikov theorem.Furthermore,the continuity of the solution map in this spaces is proved.These results can be applied directly to Novikov equation.