# Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

## Abstract

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 sp 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.

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1. 1.

This fractional Laplacian operator is sometimes called the restricted fractional Laplacian (see for example [15, 16, 49]), or regional fractional Laplacian (see for instance [43, 48]) or Dirichlet fractional Laplacian (see ).

2. 2.

The choice of the constant $$a_{N,s}$$ is motivated, among others, by the following assertion:

\begin{aligned} \lim _{s\rightarrow 0^+}(-\Delta )^s u= u\quad \text {and}\quad \lim _{s\rightarrow 1^-}(-\Delta )^s u= -\Delta u \end{aligned}

where $$\Delta$$ is the classical Laplacian. For a proof, see for instance [30, Proposition 4.4]).

3. 3.

This question was first posed by H. Brezis and J.-L. Lions in the early 1980s.

4. 4.

For the ease of the reader, let us recall that we denote by $$\Lambda _s(\Omega )$$ the best possible constant in Hardy’s inequality (see Theorem 1.1, Remark 1.2 ).

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## Acknowledgements

The authors would like to thank the anonymous reviewer for his/her careful reading of our manuscript and his/her many insightful comments and suggestions. Part of this work was realized while the first and the second authors were visiting the “Institut Elie Cartan”, Université de Lorraine. They would like to thank the Institute for the warm hospitality. B. Abdellaoui is partially supported by project MTM2016-80474-P, MINECO, Spain and by the DGRSDT, Algeria.

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