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

Abstract

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.