On the KPZ equation with fractional diffusion

Boumediene Abdellaoui, Ireneo Peral, Ana Primo

Published 2019-04-09Version 1

\begin{abstract} In this work we analyze the existence of solution 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 suppose that $f$ and $u_0$ are non negative functions satisfying some hypotheses that we will precise later. According to the value of $\a$ and the regularity of $f$, we will show the existence of a solution to problem $(P)$. \end{abstract}