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- W2751820878 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 setminusOmega), 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 $u_0$." @default.
- W2751820878 created "2017-09-15" @default.
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- W2751820878 date "2017-09-06" @default.
- W2751820878 modified "2023-09-27" @default.
- W2751820878 title "Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian" @default.
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- W2751820878 doi "https://doi.org/10.48550/arxiv.1709.01906" @default.
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