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


35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C15 Integral representations of solutions to PDEs
Full Text: DOI


[1] DOI: 10.1016/0362-546X(87)90031-9 · Zbl 0613.34013
[2] DOI: 10.1016/0362-546X(87)90111-8 · Zbl 0656.35091
[3] DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023
[4] DOI: 10.1016/0022-0396(87)90148-3 · Zbl 0625.35058
[5] DOI: 10.1016/0022-0396(85)90008-7 · Zbl 0572.34004
[6] DOI: 10.1016/0362-546X(86)90071-4 · Zbl 0611.35057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.