Journal of Integral Equations and Applications

Regularity properties of mild solutions for a class of Volterra equations with critical nonlinearities

Luciano Abadias, Edgardo Alvarez, and Carlos Lizama

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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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J. Integral Equations Applications, Volume 30, Number 2 (2018), 219-256.

First available in Project Euclid: 13 September 2018

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Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 45D05: Volterra integral equations [See also 34A12] 45N05: Abstract integral equations, integral equations in abstract spaces

Volterra integral equations local and global extension and blow up mild solutions $\epsilon $-regularity


Abadias, Luciano; Alvarez, Edgardo; Lizama, Carlos. Regularity properties of mild solutions for a class of Volterra equations with critical nonlinearities. J. Integral Equations Applications 30 (2018), no. 2, 219--256. doi:10.1216/JIE-2018-30-2-219.

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