A note on the Fujita exponent in fractional heat equation involving the Hardy potential

Boumediene Abdellaoui; Ireneo Peral; Ana Primo; Boumediene Abdellaoui; Ireneo Peral; Ana Primo

doi:10.3934/mine.2020029

Mathematics in Engineering

2020, Volume 2, Issue 4: 639-656. doi: 10.3934/mine.2020029

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A note on the Fujita exponent in fractional heat equation involving the Hardy potential

1.
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées. Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

Received: 18 November 2019 Accepted: 10 March 2020 Published: 25 May 2020

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the fractional Cauchy problem with the Hardy potential, namely, $ {u_t} + {( - \Delta )^s}u = \lambda \frac{u}{{|x{|^{2s}}}} + {u^p}{\rm{ }}\;{\rm{in}}\;{I\!\!R}^N,u(x,0) = {u_0}(x)\;{\rm{in}}\;{I\!\!R}^N, $ where \lt i \gt N \lt /i \gt \gt 2 \lt i \gt s \lt /i \gt , 0 \lt \lt i \gt s \lt /i \gt \lt 1, (-∆) \lt sup \gt \lt i \gt s \lt /i \gt \lt /sup \gt is the fractional laplacian of order 2 \lt i \gt s \lt /i \gt , \lt i \gt λ \lt /i \gt \gt 0, \lt i \gt u \lt /i \gt \lt sub \gt 0 \lt /sub \gt ≥ 0, and 1 \lt \lt i \gt p \lt /i \gt \lt \lt i \gt p \lt /i \gt \lt sub \gt + \lt /sub \gt ( \lt i \gt s \lt /i \gt , \lt i \gt λ \lt /i \gt ), where \lt i \gt p \lt /i \gt \lt sub \gt + \lt /sub \gt ( \lt i \gt λ \lt /i \gt , \lt i \gt s \lt /i \gt ) is the critical existence power to be given subsequently.
- Fujita exponent,
- fractional Cauchy heat equation with Hardy potential,
- blow-up,
- global solution
Citation: Boumediene Abdellaoui, Ireneo Peral, Ana Primo. A note on the Fujita exponent in fractional heat equation involving the Hardy potential[J]. Mathematics in Engineering, 2020, 2(4): 639-656. doi: 10.3934/mine.2020029

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Abstract

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the fractional Cauchy problem with the Hardy potential, namely, $ {u_t} + {( - \Delta )^s}u = \lambda \frac{u}{{|x{|^{2s}}}} + {u^p}{\rm{ }}\;{\rm{in}}\;{I\!\!R}^N,u(x,0) = {u_0}(x)\;{\rm{in}}\;{I\!\!R}^N, $ where \lt i \gt N \lt /i \gt \gt 2 \lt i \gt s \lt /i \gt , 0 \lt \lt i \gt s \lt /i \gt \lt 1, (-∆) \lt sup \gt \lt i \gt s \lt /i \gt \lt /sup \gt is the fractional laplacian of order 2 \lt i \gt s \lt /i \gt , \lt i \gt λ \lt /i \gt \gt 0, \lt i \gt u \lt /i \gt \lt sub \gt 0 \lt /sub \gt ≥ 0, and 1 \lt \lt i \gt p \lt /i \gt \lt \lt i \gt p \lt /i \gt \lt sub \gt + \lt /sub \gt ( \lt i \gt s \lt /i \gt , \lt i \gt λ \lt /i \gt ), where \lt i \gt p \lt /i \gt \lt sub \gt + \lt /sub \gt ( \lt i \gt λ \lt /i \gt , \lt i \gt s \lt /i \gt ) is the critical existence power to be given subsequently.

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