A semilinear integro-differential equation: global existence and hidden regularity

TL;DR

This paper proves global existence and hidden regularity for nonlinear wave equations with convolution-type integral terms, extending classical results to more complex integrodifferential equations without small initial data assumptions.

Contribution

It introduces a hidden regularity result and establishes global solutions for nonlinear integrodifferential wave equations with general nonlinearities and kernels, without smallness constraints.

Findings

Global existence of strong and mild solutions

Hidden regularity of the normal derivative trace

Extension of linear wave equation results to nonlinear integrodifferential cases

Abstract

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to state results about global existence of strong and mild solutions without any further smallness on the initial data. Then we define the trace of the normal derivative of the solution showing a regularity result. In such a way we extend to integrodifferential equations with nonlinear term well-known results available in the literature for linear wave equations with memory.