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Fractional Cauchy problem with memory effects

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  • Luciano Abadias
  • Edgardo Alvarez

Abstract

We give an original representation integral formula for the resolvent families associated to the fractional Cauchy equation with memory effects CDtαu(t)−Au(t)+∫0tβ(t−s)Au(s)ds=f(t,u(t)),t∈[0,T],T>0,where u(0)=u0∈X and A is a sectorial operator on a Banach space X. Moreover, we get spatial bounds for the resolvent families in order to study global or blow up mild solutions when the nonlinearity f satisfies a locally Lipschitz condition. The case of critical nonlinearities is also treated.

Suggested Citation

  • Luciano Abadias & Edgardo Alvarez, 2020. "Fractional Cauchy problem with memory effects," Mathematische Nachrichten, Wiley Blackwell, vol. 293(10), pages 1846-1872, October.
    • Handle: RePEc:bla:mathna:v:293:y:2020:i:10:p:1846-1872
    DOI: 10.1002/mana.201800342
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    References listed on IDEAS

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    1. Bruno Andrade & Arlúcio Viana, 2016. "Integrodifferential equations with applications to a plate equation with memory," Mathematische Nachrichten, Wiley Blackwell, vol. 289(17-18), pages 2159-2172, December.
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