Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian
2017
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 $\partial \Omega$, $n> 2s, \;s \in (0,1)$, $q>0$, ${q(2s-1) 0$. 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 $u_0$.
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