Energy estimates for the Cauchy problem of Klein–Gordon-type equations with non-effective and very fast oscillating time-dependent potential

The global existence of small self-interacting scalar field propagating in the contracting universe

“…Hirosawa and Nunes do Nascimento [15] have proved some energy estimates for Klein-Gordon equations with time-dependent potential:…”

On the wave-like energy estimates of Klein-Gordon type equations with time dependent potential

“…For M , we introduce the following properties with parameters α, β and γ: In [5], β ≥ (−γ + 3)/2 is assumed instead of (2.9) without assuming (M1), that is, only the trivial case α = −γ + 2 in Remark 2.1 (ii), is considered. The following M is an example of the non-trivial case α < −γ + 2.…”

“…A conclusion of [5] is that the same estimate from above in (1.2) as in the case δ = 0 is valid under some suitable assumptions to δ(t) which permit lim sup t→∞ (1 + t) 2 δ(t) = ∞. The main purposes of this paper is to determine the conditions for δ(t) of (1.3) with µ = 0 that the generalized modified energy conservation of wave type defined later, is established.…”

“…On the other hand, if ν > 1, that is, |M | ∈ L 1 ([0, ∞)), then the solution has more wave-like property. In [5], the following model is studied as a perturbation problem of [2]:…”

Small Data Wave Maps in Cyclic Spacetime

“…For more applications of the Floquet-Lyapunov theory to hyperbolic equations with oscillating coefficients see [14,16,20] and the bibliography therein. On the other hand, to study the hyperbolic equations with oscillating coefficients one can appeal to the so-called method of zone (see, e.g., [5,6,21,25] and the bibliography therein).…”