Strichartz estimates for harmonic potential with time-decaying coefficient

Final state problem for nonlinear Schrödinger equations with time-decaying harmonic oscillators

“…Throughout, for q ∈ [1, ∞], the norm • q denotes the norm on L q (R n ). Additionally, for q, r ∈ [1, ∞] and a < b, we define the time-weighted Bochner-Lebesgue space L q λ ((a, b); L r (R n )) (see Kawamoto-Yonenayama [10] and Kawamoto [9]) as follows:…”

“…Then, we introduce the so-called Strichartz estimate for associated propagator U 0 (t, s), which was first obtained by [10] for the case of the limited coefficients σ(t) and next under more generalized condition, including assumption 1.1, which was considered by [9].…”

Asymptotic behavior of solutions to nonlinear Schrödinger equations with time-dependent harmonic potentials

“…hold. Models of σ(t) for k = 0 (i.e., λ = 0) can be observed in, e.g., Naito [18] and Willett [23] and the models of σ(t) for λ = 0 can be observed in, e.g., Geluk-Marić-Tomić [7] (simplified models can be observed in Kawamoto [12] and Kawamoto-Yoneyama [15]).…”

“…Then, v(t, •) = U 0 (t, 0)v 0 holds. Based on the results obtained in [15] (see Korotyaev [16] and [14]), we derive the following dispersive estimate…”

“…The σ(t), which satisfies assumption 1.1 and 1.3, can be constructed with little difficulty; however, constructing the σ(t) that satisfies 1.1 and 1.4 can be very difficult. When σ(t) is non-continuous, we can construct σ(t) as presented in [15]. We summarize several models that satisfy our assumptions;…”