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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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