Open Access
2018 Regularity properties of mild solutions for a class of Volterra equations with critical nonlinearities
Luciano Abadias, Edgardo Alvarez, Carlos Lizama
J. Integral Equations Applications 30(2): 219-256 (2018). DOI: 10.1216/JIE-2018-30-2-219


We study a class of abstract nonlinear integral equations of convolution type defined on a Banach space. We prove the existence of a unique, locally mild solution and an extension property when the nonlinear term satisfies a local Lipschitz condition. Moreover, we guarantee the existence of the global mild solution and blow up profiles for a large class of kernels and nonlinearities. If the nonlinearity has critical growth, we prove the existence of the local $\epsilon $-mild solution. Our results improve and extend recent results for special classes of kernels corresponding to nonlocal in time equations. We give an example to illustrate the application of the theorems so obtained.


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Luciano Abadias. Edgardo Alvarez. Carlos Lizama. "Regularity properties of mild solutions for a class of Volterra equations with critical nonlinearities." J. Integral Equations Applications 30 (2) 219 - 256, 2018.


Published: 2018
First available in Project Euclid: 13 September 2018

zbMATH: 06979940
MathSciNet: MR3853572
Digital Object Identifier: 10.1216/JIE-2018-30-2-219

Primary: 34A12 , 45D05 , 45N05

Keywords: $\epsilon $-regularity , extension and blow up , local and global , Mild solutions , Volterra integral equations

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

Vol.30 • No. 2 • 2018
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