Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary
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Journal of Evolution Equations
Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary Boumediene Abdellaoui, Kheireddine Biroud and El- Haj Laamri
Abstract. We consider the problem ⎧ up ⎪ ⎪ ⎨ u t + (−)s u = λ 2s δ (x) (P) u(x, 0) = u 0 (x) ⎪ ⎪ ⎩ u =0
in T ≡ × (0, T ), in , in (IR N \ ) × (0, T ),
where ⊂ IR N is a bounded regular domain (in the sense that ∂ is of class C 0,1 ), δ(x) = dist(x, ∂), 0 < s < 1, p > 0, λ > 0. The purpose of this work is twofold. First We analyze the interplay between the parameters s, p and λ 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.
1. Introduction Recently, great attention has been focused on the study of nonlocal diffusion equations, i.e., first-order evolution equations driven by a nonlocal operator. These operators play a crucial role in describing several phenomena as, for instance, the thin obstacle problem, anomalous diffusion, quasi-geostrophic flows.... See, for instance, Vázquez [49], Bucur and Valdinoci [23], Molica Bisci et al. [43] and the references therein. The use of such operators reflects the need to model long-distance effects not included in the usual, local diffusion operators such as the Laplacian. Prototypical examples of a nonlocal operator are the various versions of the fractional Laplacian. Note that the proper definition of a fractional Laplacian is not obvious and offers some choices. In the literature, several definitions, not always equivalent to one another, and different terminologies are found. However, when working in the whole IR N , the alternative definitions are equivalent ; we refer the interested reader to Di Nezza et al. [30], Molica Bisci et al. [43] and Caffarelli and Silvestre [25] and the references therein. Contrary to the case of full space IR N , several different fractional Laplacians can be defined on a open subset = IR N . These alternatives correspond to different ways Mathematics Subject Classification: 35B05, 35K15, 35B40, 35K55, 35K65 Keywords: Fractional Nonlinear parabolic problems, Singular Hardy potential, Complete blow-up results.
B. Abdellaoui et al.
J. Evol. Equ.
in which the information coming from the boundary and the exterior of domain is to be taken into account. In particular, two different fractional Laplacians are widely studied in the literature, contributions include [15,16,25,43,48,49] and references therein. Needless to say, these references do not exhaust the rich literature on the subject. Here, our problem is posed on a bounded domain with homogeneous Dirichlet boundary conditions, that is u = 0 in IR N \. In other worlds, the Dirichlet datum is given in IR N \ and not simply on ∂. Then we are working with the following fractional Laplacian operator 1 which can be defined as: u(x) − u(y) dy, 0 < s < 1 (
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