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Convergence estimate for second order Cauchy problems with a small parameter. (English) Zbl 0952.35151
Extending his previous results [J. Math. Anal. Appl. 174, No. 1, 95-117 (1993; Zbl 0851.35095)] the author derives estimates for $$u_{\varepsilon} - u_0$$ and $$u_{\varepsilon}' - u_0'$$, where $$u_{\varepsilon }$$ and $$u_0$$ are the solutions of the initial value problem $\varepsilon u_{tt} + Au_t + Bu + f(u) = 0, \;u(0) = u_{0\varepsilon }, u_t (0) = u_{1\varepsilon }$ $$(\varepsilon >0)$$ and its limit $Au_t + Bu + f(u) = 0, \;u(0) = u_{00},$ respectively. Here $$A,B$$ are commuting positive selfadjoint operators in a Hilbert space, the data $$f,u_{0\varepsilon },u_{1\varepsilon }$$ are subject to some mild conditions.
Reviewer: A.Kufner (Praha)

##### MSC:
 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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##### References:
 [1] Engel, K.-J.: On singular perturbations of second order Cauchy problems. Pac. J. Math. 152 (1992), 79-91. · Zbl 0743.34063 [2] Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North Holland, 1985. · Zbl 0564.34063 [3] Najman, B.: Time singular limit of semilinear wave equations with damping. J. Math. Anal. Appl. 174 (1991), 95-117. · Zbl 0851.35095
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