Existence of mild solutions for the Hamilton-Jacobi equation with critical fractional viscosity in the Besov spaces

Abstract

We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $${\partial }_{t}u+(-\Delta {)}^{1/2}u=|\nabla u{|}^p,x\in {\mathbb{R}}^N,t>0,u(x,0)=u_0(x),x\in {\mathbb{R}}^N,$$ where $p>1$ and $u_0\in B_{r,1}^1({\mathbb{R}}^N)\cap B_{\infty ,1}^1({\mathbb{R}}^N)$ with $r\in [1,\infty ]$. We show that for sufficiently small $u_0\in {\dot{B}}_{\infty ,1}^1({\mathbb{R}}^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.

References in corpus (1)

• Global well-posedness of the critical Burgers equation in critical Besov spaces