TY - JOUR

T1 - Lower Bounds for Order of Decay or of Growthin Time for Solutions to Linear and Non-linear Schrödinger Equations

AU - Ozawa, Tohru

AU - Hayashi, Nakao

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1989

Y1 - 1989

N2 - We Study lower bounds of decay (or of growth) order in time for solutions to the Cauchy problem for the Schrodinger equation: where f is a linear or non-linear complex-valued function. Under some conditions on f and φ, it is shown that every nontrivial solution u has the estimate for sufficiently large k>0 and for any q∈[2, ∞]. In the previous work [12] of the first named author, we imposed on the assumption that u is asymptotically free. In this article, however, we shall show the assumption is, in fact, irrelevant to the results.

AB - We Study lower bounds of decay (or of growth) order in time for solutions to the Cauchy problem for the Schrodinger equation: where f is a linear or non-linear complex-valued function. Under some conditions on f and φ, it is shown that every nontrivial solution u has the estimate for sufficiently large k>0 and for any q∈[2, ∞]. In the previous work [12] of the first named author, we imposed on the assumption that u is asymptotically free. In this article, however, we shall show the assumption is, in fact, irrelevant to the results.

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U2 - 10.2977/prims/1195172508

DO - 10.2977/prims/1195172508

M3 - Article

AN - SCOPUS:85008030376

VL - 25

SP - 847

EP - 859

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

SN - 0034-5318

IS - 6

ER -