Time-dependent propagation speed vs strong damping for degenerate linear hyperbolic equations

TL;DR

This paper studies how time-dependent propagation speed and strong damping influence the well-posedness and regularity of solutions in degenerate hyperbolic equations, revealing a threshold effect based on regularity.

Contribution

It introduces a threshold effect in degenerate wave equations with time-dependent speed and strong damping, showing when damping dominates or becomes ineffective.

Findings

Damping prevails when the propagation speed is sufficiently regular.

Solutions exhibit regularizing effects similar to parabolic equations under certain conditions.

Irregular propagation speeds can nullify damping effects, making the equation behave non-dissipatively.

Abstract

We consider a degenerate abstract wave equation with a time-dependent propagation speed. We investigate the influence of a strong dissipation, namely a friction term that depends on a power of the elastic operator. We discover a threshold effect. If the propagation speed is regular enough, then the damping prevails, and therefore the initial value problem is well-posed in Sobolev spaces. Solutions also exhibit a regularizing effect analogous to parabolic problems. As expected, the stronger is the damping, the lower is the required regularity. On the contrary, if the propagation speed is not regular enough, there are examples where the damping is ineffective, and the dissipative equation behaves as the non-dissipative one.