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A singular perturbation problem for nonlinear damped hyperbolic equations. (English) Zbl 0701.35013
Let \(\beta\) be a small positive parameter. The author studies the asymptotic behavior of solutions u(\(\beta\)) of the Cauchy problem \[ (1)\quad u_{tt}(t)+Au_ t(t)+\alpha Au(t)+\beta A^ 2u(t)+G(u(t))=0,\quad u(0)=u_ 0,\quad u_ t(0)=u_ 1 \] when \(\beta\to 0\). Here \(\alpha >0\) is fixed, u: [0,\(+\infty)\to H\) is a continuous vector valued function, H is a real Hilbert space, A is a densely defined selfadjoint positive operator such that \((Av,v)\geq \zeta \| v\|^ 2\), \(\zeta >0\). Set \(W=D(A)\), \(V=D(A^{1/2})\); then \(W\subset V\subset H\) are supposed to be compactly embedded. The operator G: \(W\to H\) is the Gateaux derivative of a convex functional J: \(W\to [0,+\infty).\)
Under stronger assumptions on A and G, which we do not write, it is proved that each weak solution of (1), with \(u_ 0\in W\), \(u_ 1\in V\), decays exponentially in time, and the minimal rate of decay is the same as for the solutions of linearised equation (1), i.e. with \(G=0.\)
Sufficient conditions under which strong solutions u(\(\beta\)) of (1) converge to the unique solution of (1) with \(\beta =0\), when \(\beta\to 0\), are investigated. Some concrete realisations of (1) are given.
Reviewer: D.Huet

MSC:
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C15 Integral representations of solutions to PDEs
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[1] DOI: 10.1016/0362-546X(87)90031-9 · Zbl 0613.34013 · doi:10.1016/0362-546X(87)90031-9
[2] DOI: 10.1016/0362-546X(87)90111-8 · Zbl 0656.35091 · doi:10.1016/0362-546X(87)90111-8
[3] DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[4] DOI: 10.1016/0022-0396(87)90148-3 · Zbl 0625.35058 · doi:10.1016/0022-0396(87)90148-3
[5] DOI: 10.1016/0022-0396(85)90008-7 · Zbl 0572.34004 · doi:10.1016/0022-0396(85)90008-7
[6] DOI: 10.1016/0362-546X(86)90071-4 · Zbl 0611.35057 · doi:10.1016/0362-546X(86)90071-4
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