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, $$ \partial_t u + (-\Delta)^{ 1/2} u = |\nabla u|^p, \quad x \in \mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ 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.