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July  2009, 8(4): 1313-1332. doi: 10.3934/cpaa.2009.8.1313

## Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates

 1 Dipartimento di Matematica, University of Pisa, Italy 2 Dipartimento di Matematica Applicata, University of Pisa, Italy

Received  June 2008 Revised  November 2008 Published  March 2009

We consider the second order Cauchy problem

$\varepsilon u_\varepsilon''+ u_\varepsilon'+m(|A^{1/2}u_\varepsilon|^2)Au_\varepsilon=0, \quad u_\varepsilon(0)=u_0,\quad u_\varepsilon'(0)=u_1,$

and the first order limit problem

$u'+m(|A^{1/2}u_\varepsilon|^2)Au=0, \quad u(0)=u_0,$

where $\varepsilon>0$, $H$ is a Hilbert space, $A$ is a self-adjoint nonnegative operator on $H$ with dense domain $D(A)$, $(u_0,u_1)\in D(A^{3/2})\times D(A^{1/2})$, and $m:[0,+\infty)\to [0,+\infty)$ is a function of class $C^1$.
We prove global-in-time estimates for the difference $u_\varepsilon(t)-u(t)$ provided that $u_0$ satisfies the nondegeneracy condition $m(|A^{1/2}u_0|^2)>0$, and the function $\sigma m(\sigma^2)$ is nondecreasing in a right neighborhood of its zeroes.
The abstract results apply to parabolic and hyperbolic partial differential equations with non-local nonlinearities of Kirchhoff type.

Citation: Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313
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