Research article Special Issues

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

  • 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.

    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

    Related Papers:

  • 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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    [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 ut = ∆u + u1+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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