A singular perturbation problem for nonlinear damped hyperbolic equations. (English) Zbl 0701.35013

Biler, Piotr

Proc. R. Soc. Edinb., Sect. A 111, No. 1-2, 21-31 (1989).

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

Keywords:

singular perturbation; nonlinear damped hyperbolic equations; exponential decay; asymptotic behavior; Cauchy problem; weak solution; minimal rate of decay; strong solutions

References:

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