On the parabolic regime of a hyperbolic equation with weak dissipation: the coercive case

TL;DR

This paper investigates the decay behavior of solutions to a family of Kirchhoff hyperbolic equations with weak dissipation, establishing optimal decay estimates and revealing a weaker analogy with parabolic equations in the coercive case.

Contribution

It provides the first optimal decay and decay-error estimates for the coercive case of hyperbolic equations with weak dissipation, highlighting differences from the noncoercive case.

Findings

Solutions decay to zero as time approaches infinity.

Decay estimates are optimal and differ from noncoercive cases.

The analogy between hyperbolic and parabolic equations is weaker in the coercive case.

Abstract

We consider a family of Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t -> +infinity. It is well-known that, when the decay of the coefficient is slow enough, solutions behave as solutions of the corresponding parabolic equation, and in particular they decay to 0 as t -> +infinity. In this paper we consider the nondegenerate and coercive case, and we prove optimal decay estimates for the hyperbolic problem, and optimal decay-error estimates for the difference between solutions of the hyperbolic and the parabolic problem. These estimates show a quite surprising fact: in the coercive case the analogy between parabolic equations and dissipative hyperbolic equations is weaker than in the noncoercive case. This is actually a result for the corresponding linear equations with time-dependent coefficients. The nonlinear term comes into play only in the last step of the proof.