Lower bound estimate for the dissipative nonlinear Schrödinger equation

Upper and lower $L^2$-decay bounds for a class of derivative nonlinear Schrödinger equations

“…Indeed, it is shown in [26] that the small data solution u(t, x) to (1.3) decays like O(t −1/2 (log t) −1/2 ) in L ∞ (R x ) as t → +∞ if Im λ < 0. This gain of additional logarithmic time decay should be interpreted as another kind of longrange effect (see also [1], [2], [3], [4], [8], [9], [10], [11], [12], [13], [14], [16], [17], [18], [21], [24], [25], and so on). Time decay in L 2 -norm is also investigated by several authors.…”

“…The authors are grateful to Dr.Takuya Sato for useful conversations on his recent works [24], [25], [15], which motivate the present work. Thanks are also due to Professors Satoshi Masaki, Soichiro Katayama and Mamoru Okamoto for their helpful comments on the authors' preceding work [19].…”

Asymptotic behavior of solutions to a dissipative nonlinear Schrödinger equation with time dependent harmonic potentials

“…Hence, the L 2 -norm of the corresponding solution is monotone decreasing in t ≥ 0, however it is whether the L 2 -norm decays or not as t goes to infinity. In recent works [17], [21], [37], [45], [48], [49], [52], it is known that p = 1 + 2/n is the critical exponent to exhibit the L 2 -decay of dissipative solutions to (1.2). The L 2 -lower bound of the dissipative solution is proved when p > 1 + 2/n in [49].…”

“…In recent works [17], [21], [37], [45], [48], [49], [52], it is known that p = 1 + 2/n is the critical exponent to exhibit the L 2 -decay of dissipative solutions to (1.2). The L 2 -lower bound of the dissipative solution is proved when p > 1 + 2/n in [49]. For p ≤ 1 + 2/n, Kita-Shimomura [37] and Shimomura [52] observed the dissipative structure of (1.2) under Im λ < 0 and proved the L 2 -decay of dissipative solutions (cf.…”

Upper and lower $ L^2 $-decay bounds for a class of derivative nonlinear Schrödinger equations

“…Indeed, it is shown in [27] that the small data solution u(t, x) to (3) decays like O(t −1/2 (log t) −1/2 ) in L ∞ (R x ) as t → +∞ if Im λ < 0. This gain of additional logarithmic time decay should be interpreted as another kind of long-range effect (see also [1], [2], [3], [4], [8], [9], [10], [11], [12], [13], [14], [16], [17], [18], [22], [25], [26], and so on). Time decay in L 2 -norm is also investigated by several authors.…”