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, ut+(Δ)su=λu|x|2s+upinIRN,u(x,0)=u0(x)inIRN, 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

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  • 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, ut+(Δ)su=λu|x|2s+upinIRN,u(x,0)=u0(x)inIRN, 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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