One application of Floquet's theory to Lp–Lq estimates for hyperbolic equations with very fast oscillations

In this paper we consider the strictly hyperbolic equation u_tt?λ²(t)b²(t)Δu=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.