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.

Keywords:

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,

References

[1]	Abdellaoui B, Peral I, Primo A (2009) Influence of the Hardy potential in a semilinear heat equation. P Roy Soc Edinb A 139: 897–926. doi: 10.1017/S0308210508000152
[2]	Abdellaoui B, Medina M, Peral I, et al. (2016) Optimal results for the fractional heat equation involving the Hardy potential. Nonlinear Anal 140: 166–207. doi: 10.1016/j.na.2016.03.013
[3]	Baras P, Goldstein JA (1984) The heat equation with a singular potential. T Am Math Soc 284: 121–139. doi: 10.1090/S0002-9947-1984-0742415-3
[4]	Barrios B, Medina M, Peral I (2014) Some remarks on the solvability of non-local elliptic problems with the Hardy potential. Commun Contemp Math 16: 1–29.
[5]	Beckner W (1995) Pitt's inequality and the uncertainty principle. P Am Math Soc 123: 1897–1905.
[6]	Blumenthal RM, Getoor RK (1969) Some theorems on stable processes. T Am Math Soc 95: 263– 273.
[7]	Caffarelli L, Figalli A (2013) Regularity of solutions to the parabolic fractional obstacle problem. J Reine Angew Math 680: 191–233.
[8]	Di Nezza E, Palatucci G, Valdinoci E (2012) Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math 136: 521–573. doi: 10.1016/j.bulsci.2011.12.004
[9]	Frank R, Lieb EH, Seiringer R (2008) Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J Am Math Soc 20: 925–950.
[10]	Fujita H (1966) On the blowing up of solutions of the Cauchy problem for u_t = ∆u + u^1+α. J Fac Sci Univ Tokyo Sect I 13: 109–124.
[11]	Guedda M, Kirane M (2001) Criticality for some evolution equations. Diff Equat 37: 540–550. doi: 10.1023/A:1019283624558
[12]	Herbst IW (1977) Spectral theory of the operator (p² + m²)^1/2 - Ze²/r. Commun Math Phys 53: 285–294. doi: 10.1007/BF01609852
[13]	Kobayashi K, Sino T, Tanaka H (1977) On the growing-up problem for semilinear heat equations. J Math Soc JPN 29: 407–424. doi: 10.2969/jmsj/02930407
[14]	Landkof N (1972) Foundations of Modern Potential Theory, Springer-Verlag.
[15]	Leonori T, Peral I, Primo A, et al. (2015) Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete Cont Dyn A 35: 6031–6068. doi: 10.3934/dcds.2015.35.6031
[16]	Mitidieri E, Pohozhaev SI (2014) A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities, Proceedings of the Steklov Institute of Mathematics.
[17]	Peral I, Soria F (2021) Elliptic and Parabolic Equations involving the Hardy-Leray Potential.
[18]	Polya G (1923) On the zeros of an integral function represented by Fourier's integral. Messenger Math 52: 185–188.
[19]	Quittner P, Souplet P (2007) Superlinear Parabolic Problems Blow-up, Global Existence and Steady States, Birkhauser, Basel, Switzerland.
[20]	Riesz M (1938) Intégrales de Riemann-Liouville et potenciels. Acta Sci Math Szeged 9: 1–42.
[21]	Silvestre L (2012) On the differentiability of the solution to an equation with drift and fractional diffusion. Indiana U Math J 61: 557–584. doi: 10.1512/iumj.2012.61.4568
[22]	Sugitani S (1975) On nonexistence of global solutions for some nonlinear integral equations. Osaka J Math 12: 45–51.
[23]	Stein EM, Weiss G (1958) Fractional integrals on n-dimensional Euclidean space. J Math Mech 7: 503–514.
[24]	Weissler F (1981) Existence and nonexistence of global solutions for a semilinear heat equation. Israel Mat 38: 29–40. doi: 10.1007/BF02761845
[25]	Yafaev D (1999) Sharp constants in the Hardy-Rellich inequalities. J Funct Anal 168: 12–144.

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