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

Ghisi, Marina and Gobbino, Massimo (2008) Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates. Journal of Differential Equations, 245/20 (10). pp. 2979-3007. ISSN 0022-0396

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We consider the second order Cauchy problem $$u_\epsilon''+u_\epsilon'+m{|A^1/2 u_\epsilon|}Au_\epsilon=0, u_\epsilon(0)=u_0, u_\epsilon'(0)=u_1,$$ and the first order limit problem $$u'+m{|A^1/2u|}Au=0, u(0)=u_0,$$ where $\epsilon>0$, $H$ is a Hilbert space, $A$ is a self-adjoint nonnegative operator on $H$ with dense domain $D(A)$, and $m:[0,+\infty)\rightarrow [0,+\infty)$ is a function of class $C^{1}$. We prove decay estimates (as $t\rightarrow +\infty$) for solutions of the first order problem, and we show that analogous estimates hold true for solutions of the second order problem provided that $\epsilon$ 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 non-local nonlinearities of Kirchhoff type.