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• W2963140952 abstract "In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem (Pθ){ut+(−Δ)su=λu|x|2s+θup+cfin Ω×(0,T),u(x,t)>0in Ω×(0,T),u(x,t)=0in (RN∖Ω)×[0,T),u(x,0)=u0(x)if x∈Ω, where N>2s, 0<s<1, (−Δ)s is the fractional Laplacian of order 2s, p>1, c,λ>0, θ={0,1}, and u0,f⩾0 are in a suitable class of functions. The main results in the article are: (1) Optimal results about existence and instantaneous and complete blow up in the linear problem (P0), where the best constant in the fractional Hardy inequality, ΛN,s, provides the threshold between existence and nonexistence. To obtain local sharp estimates of the solutions it is required to prove a weak Harnack inequality for a weighted operator that appears in a natural way. (2) The existence of a critical power p+(s,λ) in the semilinear problem (P1) be such that: (a) If p>p+(s,λ), the problem has no weak positive supersolutions and a phenomenon of complete and instantaneous blow up happens. (b) If p<p+(s,λ), there exists a positive solution for a suitable class of nonnegative data." @default.
• W2963140952 created "2019-07-30" @default.
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• W2963140952 date "2016-07-01" @default.
• W2963140952 modified "2023-09-28" @default.
• W2963140952 title "Optimal results for the fractional heat equation involving the Hardy potential" @default.
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• W2963140952 doi "https://doi.org/10.1016/j.na.2016.03.013" @default.
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