The regularity properties and blow-up for convolution wave equations and applications

TL;DR

This paper investigates the existence, regularity, and blow-up phenomena of solutions to convolution wave equations with general kernels, extending understanding of nonlocal wave equations in physics through operator theory.

Contribution

It introduces a framework for analyzing both linear and nonlinear convolution wave equations with general kernels using operator functions in Banach spaces, including blow-up conditions.

Findings

Established local and global existence of solutions.

Derived regularity properties based on fractional powers of sectorial operators.

Provided conditions for finite time blow-up.

Abstract

In this paper, the Cauchy problem for linear and nonlinear convolution wave equations are studied.The equation involves convolution terms with a general kernel functions whose Fourier transform are operator functions defined in a Banach space E together with some growth conditions. Here, assuming enough smoothness on the initial data and the operator functions, the local, global existence, uniqueness and regularity properties of solutions are established in terms of fractional powers of given sectorial operator functon. Furthermore, conditions for finite time blow-up are provided. By choosing the space E and the operators, the regularity properties the wide class of nonlocal wave equations in the field of physics are obtained.