Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior

Abstract

We consider the Cauchy problem for the critical Burgers equation. The existence and the uniqueness of global solutions for small initial data are studied in the Besov space ${\dot{B}}_{\infty ,1}^0({\mathbb{R}}^n)$ and it is shown that the global solutions are bounded in time. We also study the large time behavior of the solutions with the initial data $u_0\in L^1({\mathbb{R}}^n)\cap {\dot{B}}_{\infty ,1}^0({\mathbb{R}}^n)$ to show that the solution behaves like the Poisson kernel.