Linear hyperbolic equations with time-dependent propagation speed and strong damping

TL;DR

This paper studies second order linear equations with time-dependent coefficients and strong damping, revealing a threshold effect where damping dominates or is ineffective depending on the damping exponent and regularity of the coefficient.

Contribution

It identifies a threshold at damping exponent 1/2 dictating when damping overcomes time-dependent coefficient effects, including counterexamples for optimality.

Findings

Damping dominates when exponent > 1/2, making the equation behave as if c(t) were constant.

When exponent < 1/2, the regularity of c(t) determines damping effectiveness.

Counterexamples demonstrate the sharpness of the established thresholds.

Abstract

We consider a second order linear equation with a time-dependent coefficient c(t) in front of the "elastic" operator. For these equations it is well-known that a higher space-regularity of initial data compensates a lower time-regularity of c(t). In this paper we investigate the influence of a strong dissipation, namely a friction term which depends on a power of the elastic operator. What we discover is a threshold effect. When the exponent of the elastic operator in the friction term is greater than 1/2, the damping prevails and the equation behaves as if the coefficient c(t) were constant. When the exponent is less than 1/2, the time-regularity of c(t) comes into play. If c(t) is regular enough, once again the damping prevails. On the contrary, when c(t) is not regular enough the damping might be ineffective, and there are examples in which the dissipative equation behaves as the non-dissipative one. As expected, the stronger is the damping, the lower is the time-regularity threshold. We also provide counterexamples showing the optimality of our results.