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One application of Floquet's theory to L p – L q estimates for hyperbolic equations with very fast oscillations
Author(s) -
Reissig Michael,
Yagdjian Karen
Publication year - 1999
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19990725)22:11<937::aid-mma28>3.0.co;2-o
Subject(s) - mathematics , floquet theory , constant (computer programming) , function (biology) , mathematical analysis , class (philosophy) , hyperbolic partial differential equation , cauchy problem , energy (signal processing) , energy method , initial value problem , differential equation , nonlinear system , statistics , physics , quantum mechanics , evolutionary biology , artificial intelligence , computer science , biology , programming language
In this paper we consider the strictly hyperbolic equation u t t − λ 2 ( t ) b 2 ( t )Δ u =0. The coefficient consists of an increasing function λ = λ ( t ) and a non‐constant periodic function b = b ( t ). We study the question for the influence of these parts on L p – L q decay estimates for the solution of the Cauchy problem. A fairly wide class of equations will be described for which the influence of the oscillating part dominates. This implies, on the one hand, that there exist no L p – L q decay estimates and, on the other hand, that the energy estimate from Gronwall's inequality is near to an optimal one. Copyright © 1999 John Wiley & Sons, Ltd.