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Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation. (English) Zbl 1255.35027

The author considers the Kirchhoff equations with a small parameter \[ \epsilon u''_\epsilon (t) +(1+t)^{-p} u'_\epsilon (t) +|A^{1/2}u_\epsilon|^{2\gamma} A u_\epsilon (t) =0\tag{1} \] with initial conditions, where \(A\) is a self-adjoint linear operator on a Hilbert space \(H\) with dense domain \(D(A)\), \(|\cdot|\) is the norm in \(H\), \(0\leq p\leq 1\), \(\gamma\geq 1\).
By exploiting the known properties for the Kirchhoff equations and employing careful energy estimates, the author proves the existence of global solutions provided that \(\epsilon\) is small with respect to the size of initial data. Moreover, the global-in-time error estimates on \(u_\epsilon -u\) in terms of some power of \(\epsilon\) are established, where \(u\) is the solution of the limiting problem obtained formally by setting \(\epsilon =0\) in (1).

MSC:

35B25 Singular perturbations in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L90 Abstract hyperbolic equations
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