Asymptotic behavior of solutions to the short pulse equation with critical nonlinearity

We consider the Cauchy problem for the short pulse equation {utx=u+(u3)xx,x∈R,t>0,u(0,x)=u0(x),x∈R, where u0u0 is a real valued function. We prove the global existence of small solutions to the short pulse equation. Moreover we give the L∞ time decay estimate ‖u(t)‖L∞≤C(1+t)−1/2 and the asymptotic behavior of solutions.