On the KPZ equation with fractional diffusion: global regularity and existence results

Abstract

In this work we analyze the existence of solutions to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminus\Omega)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \inn\Omega,\\ \end{array}\right. $$ where $\Omega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s$ and $\frac{1}{2}<s<1$. We will assume that $f$ and $u_0$ are non negative functions satisfying some additional hypotheses that will be specified later on. Assuming certain regularity on $f$, we will prove the existence of a solution to problem $(P)$ for values $\alpha<\dfrac{s}{1-s}$, as well as the non existence of such a solution when $\alpha>\dfrac{1}{1-s}$. This behavior clearly exhibits a deep difference with the local case.