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

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 + (-Δ)^s u &= u^{-q} + f(x,u), \;u >0\; \text{in}\; (0,T) \times Ω, u &= 0 \; \mbox{in}\; (0,T) \times (\mb R^n \setminusΩ), \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 $(x,y)\in Ω\times \mb R^+ \mapsto f(x,y)$ is a bounded below Carathéodary 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}<λ_1^s(Ω)}, \end{equation} where $\la_1^s(Ω)$ 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 $u_0$.

31 pages