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Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel


Let A be a densely defined closed, linear \(\omega\)-sectorial operator of angle \(\theta \in [0,\frac{\pi }{2})\) on a Banach space X, for some \(\omega \in \mathbb {R}\). We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equation with memory: \(\displaystyle u'(t)=Au(t)+(\kappa *Au)(t), \, t >0\), \(u(0)=u_0\), associated with the (possible) singular kernel \(\kappa (t)=\alpha e^{-\beta t}\frac{t^{\mu -1}}{\Gamma (\mu )},\;\;t>0\), where \(\alpha \in \mathbb {R}\), \(\alpha \ne 0\), \(\beta \ge 0\) and \(0<\mu < 1\).

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This research has been partially supported by Conicyt, Project MEC #80180010. The work of M. Warma is partially supported by Air Force Office of Scientific Research (AFOSR) under Award NO [FA9550-18-1-0242] and Army Research Office (ARO) under Award NO: W911NF-20-1-0115.

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Correspondence to Rodrigo Ponce.

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Communicated by Abdelaziz Rhandi.

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Ponce, R., Warma, M. Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel. Semigroup Forum 102, 250–273 (2021).

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  • Volterra kind of equation
  • Generalized Mittag-Leffler functions
  • Explicit representation of solutions
  • Exponential stability of solutions
  • Heat equation with memory