Journal of Evolution Equations ( IF 1.1 ) Pub Date : 2020-10-15 , DOI: 10.1007/s00028-020-00623-9 Boumediene Abdellaoui , Kheireddine Biroud , El-Haj Laamri
We consider the problem
$$\begin{aligned} (P)\left\{ \begin{array}{llll} u_t+(-\Delta )^{s} u &{}=&{} \lambda \dfrac{u^p}{\delta ^{2s}(x)} &{} \quad \text { in }\Omega _{T}\equiv \Omega \times (0,T) , \\ u(x,0)&{}=&{}u_0(x) &{} \quad \text { in }\Omega , \\ u&{}=&{}0 &{}\quad \text { in } ({I\!\!R}^N\setminus \Omega ) \times (0,T), \end{array}\right. \end{aligned}$$where \(\Omega \subset {I\!\!R}^N\) is a bounded regular domain (in the sense that \(\partial \Omega \) is of class \({\mathcal {C}}^{0,1}\)), \(\delta (x)=\text {dist}(x,\partial \Omega )\), \(0<s<1\), \(p>0\), \(\lambda >0\). The purpose of this work is twofold. First We analyze the interplay between the parameters s, p and \(\lambda \) in order to prove the existence or the nonexistence of solution to problem (P) in a suitable sense. This extends previous similar results obtained in the local case \(s=1\). Second We will especially point out the differences between the local and nonlocal cases.
中文翻译：
边界处奇异权重的分数阶抛物问题正解的存在与不存在
我们考虑这个问题
$$ \ begin {aligned}（P）\ left \ {\ begin {array} {llll} u_t +（-\ Delta）^ {s} u＆{} =＆{} \ lambda \ dfrac {u ^ p} { \ delta ^ {2s}（x）}＆{} \ quad \ text {in} \ Omega _ {T} \ equiv \ Omega \ times（0，T），\\ u（x，0）＆{} = ＆{} u_0（x）＆{} \ quad \ text {in} \ Omega，\\ u＆{} =＆{} 0＆{} \ quad \ text {in}（{I \！\！R} ^ N \ setminus \ Omega）\ times（0，T），\ end {array} \ right。\ end {aligned} $$其中\（\ Omega \ subset {I \！\！R} ^ N \）是有界的正则域（在某种意义上，\（\ partial \ Omega \）属于\（{\ mathcal {C}} ^类{0,1} \）），\（\ delta（x）= \文本{dist}（x，\ partial \ Omega）\），\（0 <s <1 \），\（p> 0 \），\（\ lambda> 0 \）。这项工作的目的是双重的。首先，我们分析参数s， p和\（\ lambda \）之间的相互作用，以在适当的意义上证明问题（P）解的存在或不存在。这扩展了先前在本地情况下获得的类似结果\（s = 1 \）。其次，我们将特别指出本地案例与非本地案例之间的差异。
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