Critical counterexamples for linear wave equations with time-dependent propagation speed

TL;DR

This paper explores the limits of well-posedness for wave equations with time-dependent speeds, providing new counterexamples that demonstrate ill-posedness in critical parameter regimes.

Contribution

It introduces novel counterexamples for critical cases of wave equations with variable speeds, using a new approach based on Baire category theorem.

Findings

Critical cases are ill-posed with severe derivative loss.

Counterexamples show pathological behavior in certain parameter regimes.

New construction method for counterexamples using Baire category theorem.

Abstract

We investigate an abstract wave equation with a time-dependent propagation speed, and we consider both the non-dissipative case, and the case with a strong damping that depends on a power of the elastic operator. Previous results show that, depending on the values of the parameters and on the time regularity of the propagation speed, this equation exhibits either well-posedness in Sobolev spaces, or well-posedness in Gevrey spaces, or ill-posedness with severe derivative loss. In this paper we examine some critical cases that were left open by the previous literature, and we show that they fall into the pathological regime. The construction of the counterexamples requires a redesign from scratch of the basic ingredients, and a suitable application of Baire category theorem in place of the usual iteration scheme.