Hyperbolic–parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates

We consider the second order Cauchy problemεuε″+uε′+m(|A1/2uε|2)Auε=0,uε(0)=u0,uε′(0)=u1, and the first order limit problemu′+m(|A1/2u|2)Au=0,u(0)=u0, where ε>0ε>0, H is a Hilbert space, A is a self-adjoint nonnegative operator on H   with dense domain D(A)D(A), (u0,u1)∈D(A)×D(A1/2)(u0,u1)∈D(A)×D(A1/2), and m:[0,+∞)→[0,+∞) is a function of class C1C1.We prove decay estimates (as t→+∞t→+∞) for solutions of the first order problem, and we show that analogous estimates hold true for solutions of the second order problem provided that ε is small enough. We also show that our decay rates are optimal in many cases.The abstract results apply to parabolic and hyperbolic partial differential equations with nonlocal nonlinearities of Kirchhoff type.