arXiv 1709.019062017-09-06math.AP

一类含分数阶Laplacian算子的奇异抛物型方程解的存在性和稳定性

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

J. Giacomoni, Tuhina Mukherjee, K. Sreenadh

备注：: 31 pages

摘要：

本文研究了具有奇异非线性项的分数阶Laplacian的抛物型方程\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*}，其中\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*}是$\mb{R}^n$中的有界区域，具有光滑边界$\partial Ω$，$n> 2s, \;s \in (0,1)$，$q>0$，${q(2s-1)<(2s+1)}$，$u_0 \in L^\infty(Ω)\cap X_0(Ω)$和$T>0$ .我们假设映射$(x,y)\in Ω\times \mb R^+ \mapsto f(x,y)$是一个关于第二变量的局部Lipschitz的下有界Carathéoeven函数，并且对于$x \in Ω$一致地满足\begin{equation}\label{cond_on_f} { \limsup_{y \to +\infty} \frac{f(x,y)}{y}<λ_1^s(Ω)}, \end{equation}其中$\la_1^s(Ω)$是$(-Δ)^s$在$Ω$中的第一特征值，在$\mathbb{R}^n \setminus Ω$中具有齐次Dirichlet边界条件.在适当的锥条件下，证明了$(P_t^s)$弱解的存在唯一性。我们采用隐式Euler方法在时间上进行半离散，并研究了平稳问题来证明我们的结果。当我们正则化我们的初始函数$u_0$时，我们还显示了$(P_t^s)$的解的额外正则性。

英文摘要：

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 $Ω$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial Ω$, $n> 2s, \;s \in (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 \in L^\infty(Ω)\cap X_0(Ω)$ 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 $x \in Ω$ 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 $(-Δ)^s$ in $Ω$ with homogeneous Dirichlet boundary condition in $\mathbb{R}^n \setminus Ω$. 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 $u_0$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文