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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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