Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity

$\begin{equation*} \quad (P_{t}^s) \left\{\begin{split} \quad u_t + (-\Delta)^s u & = u^{-q} + f(x,u), \;u >0\; \text{in}\;(0,T) \times \Omega, \\ u & = 0 \; \mbox{in}\; (0,T) \times (\mathbb{R}^n \setminus \Omega ),\\ \quad \quad \quad \quad u(0,x)& = u_0(x) \; \mbox{in} \; {\mathbb{R}^n},\end{split}\quad \right.\end{equation*}$

where $\Omega $ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega $, $n> 2s, \;s ∈ (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 ∈ L^∞(\Omega )\cap X_0(\Omega )$ and $T>0$. We suppose that the map $(x,y)∈ \Omega × \mathbb{R}^+ \mapsto f(x,y)$ is a bounded from below Carathéodary function, locally Lipschitz with respect to the second variable and uniformly for $x ∈ \Omega $ and it satisfies

$ \begin{equation}\label{cond_on_f}{ \limsup\limits_{y \to +\infty} \frac{f(x,y)}{y}<\lambda_1^s(\Omega)}, \end{equation}$

where $\lambda_1^s(\Omega )$ is the first eigenvalue of $(-\Delta )^s$ in $\Omega $ with homogeneous Dirichlet boundary condition in $\mathbb{R}^n \setminus \Omega $. We prove the existence and uniqueness of a weak solution to $(P_t^s)$ on assuming $u_0$ satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results.We also show additional regularity on the solution of $(P_t^s)$ when we regularize our initial function $u_0$.