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

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  • 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.

    Mathematics Subject Classification: 35B25, 35B40, 35L80.


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