Hyperbolic-parabolic singular perturbation for Kirchhoff equations with weak dissipation

2009/12/18 by Ghisi, Marina, Gobbino, Massimo
#35B25 #35B40 #35L70 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0912.3694

Abstract

We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the values of the parameters.

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