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Fractional heat equation involving Hardy-Leray Potential

Boumediene Abdellaoui, Giovanni Siclari, Ana Primo

Abstract

In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$ \begin{cases} (w_t-Δw)^s=\fracλ{|x|^{2s}} w+w^p +f, &\text{ in }\mathbb{R}^N\times (0,+\infty),\\ w(x,t)=0, &\text{ in }\mathbb{R}^N\times (-\infty,0], \end{cases} $$ where $N> 2s$, $0<s<1$ and $0<λ<Λ_{N,s}$, the optimal constant in the fractional Hardy-Leray inequality. In particular we show the existence of a critical existence exponent $p_{+}(λ, s)$ and of a Fujita-type exponent $F(λ,s)$ such that the following holds: - Let $p>p_+(λ,s)$. Then there are not any non-negative supersolutions. - Let $p<p_+(λ,s)$. Then there exist local solutions while concerning global solutions we need to distinguish two cases: - Let $ 1< p\le F(λ,s)$. Here we show that a weighted norm of any positive solution blows up in finite time. - Let $F(λ,s)<p<p_+(λ,s)$. Here we prove the existence of global solutions under suitable hypotheses.

Fractional heat equation involving Hardy-Leray Potential

Abstract

In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem where , and , the optimal constant in the fractional Hardy-Leray inequality. In particular we show the existence of a critical existence exponent and of a Fujita-type exponent such that the following holds: - Let . Then there are not any non-negative supersolutions. - Let . Then there exist local solutions while concerning global solutions we need to distinguish two cases: - Let . Here we show that a weighted norm of any positive solution blows up in finite time. - Let . Here we prove the existence of global solutions under suitable hypotheses.
Paper Structure (15 sections, 28 theorems, 191 equations, 1 figure)

This paper contains 15 sections, 28 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional setting
  4. Properties of the operators $J_s$ and $H^s$
  5. Ground state transformation for the operator $H^s-\frac{\lambda}{|x|^{2s}}$.
  6. A comparison principle
  7. An extension result
  8. A comparison principle
  9. An existence result
  10. Local behaviour of weak solutions
  11. Non-existence result and Fujita-type behaviour
  12. A non-existence result
  13. A blow-up result
  14. Existence of supersolutions
  15. Open problems and subjects of further investigation

Key Result

Theorem 1.1

For all $\varphi\in \mathcal{C}^{\infty}_{0}(\mathbb{R}^n)$, we have with The constant $\Lambda_{N,s}$ is optimal and is not attained. Moreover, $\Lambda_{N,s}\to \Lambda_{N,1}:=\left(\dfrac{N-2}{2}\right)^2$, the classical Hardy constant, when $s$ tends to $1$.

Figures (1)

  • Figure 1: Fujita exponent for the fractional heat equation with Hardy potential.

Theorems & Definitions (56)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 46 more