一类Rosenau方程的Sobolev指数

Sobolev Exponent of a Generalized Rosenau Equation

在研究紧离散动力系统时,为了克服KdV方程不能描绘波与波、波与墙的相互作用而提出了Rosenau方程.主要研究如下一类Rosenau方程Cauchy问题的整体解{utt-2aΔut+Δ2utt=-bΔ2u+Δu+Δ(up),u(x,0)=ε2Φ(x),ut(x,0)=ε2ψ(x),其中,x∈Rn,n≥2,t>0,a、b是正常数,ε>0是小参数,p≥2是正整数.当b-a2>0时,运用Fourier变换和扰动方法,将在Sobolev空间中得到上面问题整体解的存在唯一性及形式解的长时间渐近性,并得到了方程的Sobolev指数是n/2-1/p-1.

In the study of the dynamics of dense discrete systems,the concept of Rosenau equation was proposed in order to overcome the shortcoming that the interactions of wave-wave and wave-wall cannot be described by the KdV equation. This paper deals with the global solution of the Cauchy problem for the following Rosenau equation{utt-2aΔut＋Δ^2utt=-bΔ^2u＋Δu＋Δ（u^p）,u（x,0）=ε^2Φ（x）,ut（x,0）=ε^2ψ（x）,where x∈Rn,n≥2,t 〉0,a and b are positive constants,ε 〉0 is a small positive parameter,p≥2 is a positive interger. For the caseb-a2〉 0,making use of Fourier transformation and perturbation method,the existence and uniqueness of the global solution for the above Rosenau equation are established in Sobolev space. The long time asymptotic behavior of the formal approximation solution isobtained. It is proved that the Sobolev exponent of the equation isn/2-1/p-1.