The Cauchy problem for the Rosenau equation

This paper considers the existence and the uniqueness of the solution for the Cauchy problem of the Rosenau equation utt+ux4+ux4tt−γuxx=f(u)xx.utt+ux4+ux4tt−γuxx=f(u)xx. We prove the existence and the uniqueness of the global solution under some conditions of f(u)f(u) and give the condition for nonexistence of a global solution. For f(u)=−β|u|puf(u)=−β|u|pu with β>0β>0 and p>0p>0, the global existence and finite-time blow-up for the problem are proved with the aid of the potential well method.