Non-uniform Dependence On Initial Data For The Modified\mu-Camassa-Holm Equation

In this thesis,we mainly consider the non-uniformly continuous dependence on initial data of the solutions to the Cauchy problem associated with a non-linear evolution equation.As wc all know,the initial data has close relation with corresponding solutions for the Cauchy problem of an evolution equation,but the factor to describe their relations is the property of the solution map u0? u(t)in the related space.Especially the local wcll-poscdncss results show that the solutions depend continuously on initial data,that is,the solution map u0? u(t)in the corresponding space is continuous,but the propcerty which wc study profoundly reveals that the solutions depend non-uniformly continuously on initial data,that is,the solution map u0?u(t)is not non-uniformly contin-uous.To prove this property,the method of Approximate solutions is the most commonly used by which several researchers have proved this property of many equations in a similar style.In our paper,based on the local wcll-posedness re-sults for this problem,we show that the solution map u0?u(t)of the modified?-Camassa-Holm equation is not uniformly continuous in Sobolev spaces HS(S)for s>5/2 on the circle.This paper is organized as follows:At first,we briefly describe the purpose and significancce of the non-uniformly continuous dependence on initial data and the results with respect to many typical nonlinear evolution equations both at home and abroad which the researchers have obtained.Secondly,we illustrate related definitions,notations of the non-uniformly continuous dependence on initial data,and then give the theorems and lemmas which are necessary in our proof.At last,we establish the non-uniformly continuous dependence on initial data of the solutions to the Cauchy problem associated with the modified ?-Camassa-Holm equation.