GLOBAL ASYMPTOTICS TOWARD THE RAREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

摘要

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation with the initial data u(0,x) = u0(x)→±, as x→±∞. (Ⅰ) Here, u- 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u- (x) - U(0,x) ∈H1(R) and u- < u+, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends to the rarefaction wave uR(x/t) as t→+∞ in the maximum norm. The proof is given by an elementary energy method.