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

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