On hyperbolic equations and systems with non-regular time dependent coefficients

TL;DR

This paper investigates higher order weakly hyperbolic equations with non-regular, bounded coefficients, introducing a 'very weak solution' concept through regularisation and demonstrating solvability for related systems.

Contribution

It extends the theory of hyperbolic equations to include non-regular coefficients by defining very weak solutions and proving their existence and convergence to classical solutions.

Findings

Existence of very weak solutions for higher order hyperbolic equations with bounded coefficients.

Recovery of classical solutions as limits of very weak solutions.

Solvability of first order hyperbolic systems with non-regular coefficients in the very weak sense.

Abstract

In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than H\"older, namely bounded coefficients. As for second order equations in \cite{GR:14} we prove that such equations admit a `very weak solution' adapted to the type of solutions that exist for regular coefficients. The main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter. Classical solutions are recovered as limit of very weak solutions. Finally, by using a reduction to block Sylvester form we conclude that any first order hyperbolic system with non-regular coefficients is solvable in the very weak sense.