Maximum principle for the weak solutions of the Cauchy problem for the fourth-order hyperbolic equations

TL;DR

This work addresses the problem of a maximum principle for the Cauchy problem of a $4^{th}$-order hyperbolic PDE with constant coefficients in a plane domain bounded by characteristics. It develops a robust operator-theoretic framework using $L$, $L^+$, $D(L)$, and $L$-traces $L_{(p)}u$, and formulates the weak solution in terms of boundary data on $\Gamma_0$ within admissible characteristic domains. The main contribution is Theorem 1, a maximum principle for weak solutions on a characteristic polygon, proven by an integration-by-parts argument that leverages boundary trace terms and is then extended from smooth to $L^2$ solutions via density, with a discussion of weak $L^2$ variants. This result provides a foundational tool for uniqueness, existence, and qualitative analysis of higher-order hyperbolic equations and their physical models, where classical traces may fail and $L$-traces govern boundary behavior.

Abstract

We investigate the maximum principle for the weak solutions to the Cauchy problem for the hyperbolic fourth-order linear equations with constant complex coefficients in the plane bounded domain