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Note on lower bounds of energy growth for solutions to wave equations. (English) Zbl 1273.35069

The model is a compact in space perturbation of the wave equation. The solution operator is denoted with \(R(t,0)\) and an upper bound on the growth of its norm is shown. Further, assuming the existence of some special null bicharacteristics, lower bounds for the norm of \(R(t,0)\) are proved. The obtained result is applied to wave equations constructed by means of techniques developed in [F. Colombini and J. Rauch, J. Reine Angew. Math. 616, 1–14 (2008; Zbl 1158.35012)].

MSC:

35B45 A priori estimates in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1158.35012
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Full Text: Euclid

References:

[1] F. Colombini and J. Rauch: Smooth localized parametric resonance for wave equations , J. Reine Angew. Math. 616 (2008), 1-14. · Zbl 1158.35012
[2] F. Colombini and S. Spagnolo: Hyperbolic equations with coefficients rapidly oscillating in time: a result of nonstability , J. Differential Equations 52 (1984), 24-38. · Zbl 0589.35073
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