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

We consider the second order Cauchy problem epsilon u ''(epsilon) +u'(epsilon) +m(A(1/2)u(epsilon)|(2))Au-epsilon = 0, u(epsilon)(0) = u0, u'(epsilon)(0) =u1, and the first order limit problem u' + m (|A(1/2)u|(2)) Au=0, u(0) = u0, where epsilon > 0, H is a Hilbert space, A is a self-adjoint nonnegative operator on H with dense domain D(A), (u(0), u(1)) is an element of D(A) x D(a(1/2)), and m:|0, +infinity)-> (0,+infinity) is a function of class C-1. We prove decay estimates (as t ->+infinity) for solutions of the first order problem, and we show that analogous estimates hold true for solutions of the second order problem provided that F 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. (C) 2008 Elsevier Inc. All rights reserved.