Optimal results for the fractional heat equation involving the Hardy potential

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• 作者：Boumediene Abdellaoui^a ; ^{boumediene.abdellaoui@uam.es" class="auth_mail" title="E-mail the corresponding author} ; Marí ; a Medina^b ; ^{maria.medina@uam.es" class="auth_mail" title="E-mail the corresponding author} ; Ireneo Peral^b ; ^{ireneo.peral@uam.es" class="auth_mail" title="E-mail the corresponding author} ; Ana Primo^b ; ^{ana.primo@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B25 ; 35K58 ; 35B33 ; 47G20
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：140
• 期：Complete
• 页码：166-207
• 全文大小：843 K

文摘

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem where N>2s$N>2s$, 0<s<1$0<s<1$, (−Δ)^s${(-\Delta )}^s$ is the fractional Laplacian of order 2s$2s$, p>1$p>1$, c,λ>0$c,\lambda >0$, θ={0,1}$\theta =\{0,1\}$, and $u_0,f⩾0$ 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)$\left(P_0\right)$, where the best constant in the fractional Hardy inequality, Λ_N,s${\Lambda }_{N,s}$, 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)$p_{+}(s,\lambda )$ in the semilinear problem (P₁)$\left(P_1\right)$ be such that:

(a) If p>p₊(s,λ)$p>p_{+}(s,\lambda )$, the problem has no weak positive supersolutions and a phenomenon of complete and instantaneous blow up happens.

(b) If p<p₊(s,λ)$p<p_{+}(s,\lambda )$, there exists a positive solution for a suitable class of nonnegative data.