Fractional heat equation with singular nonlinearity

Abstract

Our aim in this paper is to analyze the existence and regularity of solutions for the following nonlocal parabolic problem involving the fractional Laplacian with singular nonlinearity

$$\begin{aligned} \left\{ \begin{array}{rlll} u_t+(-\Delta )^{s} u &{}=\frac{f(x,t)}{u^{\gamma }} &{} \text{ in } \Omega _T:=\Omega \times (0,T),\\ u &{}=0 &{} \text{ in } ({\mathbb {R}}^{N}\backslash \Omega )\times (0,T),\\ u(\cdot ,0)&{}=u_0(\cdot ) &{} \text{ in } \Omega , \end{array} \right. \end{aligned}$$

where $\Omega$ is a bounded domain of class ${\mathcal {C}}^{0,1}$ in ${\mathbb {R}}^{N}$, $N>2s$ with $s\in (0,1)$, $\gamma >0$, $0<T<+\infty$, $f\ge 0$, $f\in L^{m}(\Omega _T)$, $m\ge 1$, is a non-negative function on $\Omega _T$ and $u_0$ is a non-negative function defined on $\Omega$ belonging to some Lebesgue spaces. The existence and regularity of the very weak and weak solutions are obtained under different assumptions on the summability of the data and on $\gamma$. One of the difficulties arising in the problem is the proof of the strict positivity of the weak solutions inside the parabolic cylinders. This proof involves the smallest eigenvalue of the fractional Laplacian and the weak comparison principle. We also deal with the case of singular Radon measures data and the asymptotic behavior of the very weak solutions.