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

TL;DR

This paper establishes existence, uniqueness, and regularity results for a singular parabolic equation involving the fractional Laplacian, using semi-discretization and stationary problem analysis.

Contribution

It introduces new existence and uniqueness results for a fractional parabolic PDE with singular nonlinearity, extending previous work to include regularity and stabilization analysis.

Findings

Proved existence and uniqueness of weak solutions.

Demonstrated additional regularity under initial data regularization.

Analyzed the stationary problem for stabilization insights.

Abstract

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 (\mb R^n \setminus\Omega), \quad \quad \quad \quad u(0,x)&=u_0(x) \; \mbox{in} \; {\mb R^n}, \end{split} \quad \right. \end{equation*} where $\Omega$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Omega$, $n> 2s, \;s \in (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 \in L^\infty(\Omega)\cap X_0(\Omega)$ and $T>0$. We suppose that the map $(x,y)\in \Omega \times \mb R^+ \mapsto f(x,y)$ is a bounded below Carath\'eodary function, locally Lipschitz with respect to second variable and uniformly for $x \in \Omega$ it satisfies \begin{equation}\label{cond_on_f} { \limsup_{y \to +\infty} \frac{f(x,y)}{y}<\lambda_1^s(\Omega)}, \end{equation} where $\la_1^s(\Omega)$ 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 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 $u_0$.