The Cauchy Problem For A Class Of Rosenau Equation

In the study of the dynamics of dense discrete systems, the case wave-wave and wave-wall interactions cannot be described using the well-known Kd V equation. To overcome this shortcoming of the Kd V equation, Rosenau proposed the so-called Rosenau equation. In this paper, we study the existence and uniqueness of local and global solutions for the Cauchy problem to the class of Rosenau equation, the main results include the following aspects:Firstly, we introduce some physical background knowledges of the Rosenau equation, and then summarize some related works and our main results.Secondly, we deals with the global solution of the Cauchy problem for a class of generalized Rosenau equation. In some case, making use of Fourier transformation and peturbation method, the well-posedness for the Rosenau equation is established in Sobolev space. The long time asymptotic behavior of the formal approximation solution is obtained. Further, we discuss the Sobolev exponent of the equation.Lastly, we consider the blow-up, the global existence of small amplitude solutions and nonlinear scattering for a class of generalized Rosenau equation in one dimensional space. First of all, using the contraction mapping theorem, we obtain the existence of the local solution. Secondly, under some assumptions about energy, we obtain the blowup result by establishing a differential inequality for a functional of the solution.Finally,we study global small-amplitude solution to this problem and their nonlinear scattering by using the contraction mapping theorem and utilizing an estimate for the uniform decay of solutions of the linearized version. Then the results are proved.