Asymptotic behavior for solutions to the dissipative nonlinear Schrödinger equations with the fractional Sobolev space

Abstract

We study the Cauchy problem for the nonlinear Schrödinger equations with dissipative nonlinearity. In particular, we show the time decay estimate for global solutions in Lebesgue space of square integrable functions. Main results of this paper are the extension of results that have been proved in the work of Hayashi et al. [Adv. Math. Phys. 5, 3702738 (2016)].

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.…”

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].…”

“…Hayashi-Li-Naumkin [17] obtained the L 2 -decay rate of the dissipative solution under the condition (1.4). Hoshino [21] showed the L 2 -decay order of dissipative solutions in the Sobolev space with a low differential order. Ogawa-sato [45] and Sato [48] showed the L 2 -decay order of solutions which has analytic or Gevrey regularity in spatial variable.…”

On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

“…A combination of suitable iteration methods, maximum principle and method of moving planes, is useful to detect symmetries of positive solutions and nonexistence results (see, for example, [3]). We also mention the recent works of Peng-Zhao [4] (global existence and blow-up of solutions) and Hoshino [5] (asymptotic behavior of solutions).…”