## 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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### References:

 [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
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