Optimal results for the fractional heat equation involving the Hardy potential

By Boumediene Abdellaoui, María Medina, I. Peral, A. Primo. 2014

51 citations recorded by the source corpus.

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c f\mbox{ in } \Omega\times (0,T),\\ u(x,t)&>&0\inn \Omega\times (0,T),\\ u(x,t)&=&0\inn (\ren\setminus\Omega)\times[ 0,T),\\ u(x,0)&=&u_0(x) \mbox{ if }x\inO, \end{array} \right. $$ where $N> 2s$, $0 1$, $c,l>0$, $u_0\ge 0$, $f\ge 0$ are in a suitable class of functions and $\theta=\{0,1\}$. Notice that $(P_0)$ is a linear problem, while $(P_1)$ is a semilinear problem. The main features in the article are: \begin{enumerate} \item Optimal results about \emph{existence} and \emph{instantaneous and complete blow up} in the linear problem $(P_0)$, where the best constant $\Lambda_{N,s}$ in the fractional Hardy inequality provides the threshold between existence and nonexistence. Similar results in the local heat equation were obtained by Baras and Goldstein in \cite{BaGo}. However, in the fractional setting the arguments are much more involved and they require the proof of a weak Harnack inequality for a weighted operator that appear in a natural way. Once this Harnack inequality is obtained, the optimal results follow as a simpler consequence than in the classical case. \item The existence of a critical power $p_+(s,\lambda)$ in the semilinear problem $(P_1)$ such that: \begin{enumerate} \item If $p> p_+(s,\lambda)$, the problem has no weak positive supersolutions and a phenomenon of \emph{complete and instantaneous blow up} happens. \item If $p< p_+(s,\lambda)$, there exists a positive solution for a suitable class of nonnegative data. \end{enumerate} \end{enumerate}

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/5164cc0da498f50f2be6d0ab3a4aa7708d6691d9

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In Applied Mathematics

Citations in the corpus

On fractional p-Laplacian parabolic problem with general data (2016, background): “The case with Hardy potential and under ”natural” condition on (f, u0) has been studied in [2].”
On the fractional p-Laplacian equations with weight and general datum (2016, methodology): “e Br(x0) = Br(0). For simplicity of typing, we will write Br in place of Br(0). We will use systematically the next weighted version of the Poincar´e-Wirtinger inequality. We refer to the Appendix of [3] for the proof. Theorem 5.3. Let w∈ Ws,p β (B2) and assume that ψis a radial decreasing function ”
Fractional Laplacian with Hardy potential (2017, background): “It is proved by Abdellaoui, Medina, Peral and Primo [2, 1] that for κ > κ∗ the operator has no weak positive supersolutions and the phenomenon of complete and instantaneous blow-up occurs, whereas for κ ≤ κ∗ nontrivial nonnegative solutions of (1.”