Generalized energy conservation for klein–gordon type equations
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The aim of this paper is to derive energy estimates for solutions of the Cauchy problem for the Klein–Gordon type equation u_(tt) - Δu + m(t)^2u = 0. The coefficient m is given by m(t)^2 = λ(t)^2 + p(t) with a decreasing, smooth shape function λ and an oscillating, smooth and bounded perturbation function p. We study under which assumptions for λ and p one can expect results about a generalization of energy conservation. The main theorems of this note deal with m belonging to C^M, M ≥ 2, and m belonging to the Gevrey class γ^(s), s ≥ 1.
- Osaka Journal of Mathematics
Osaka Journal of Mathematics 49(2), 297-323, 2012-06
Osaka University and Osaka City University, Departments of Mathematics