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Abstract

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., $λ | u |^{2} u$ . Our aim is to obtain the two results. One asserts that, if the $L^{2}$ -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ${(log t)}^{- 1 / 2}$ , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ${(log t)}^{- 1 / 2}$ in $L^{2}$ .

Keywords

Nonlinear Schrödinger equation decay estimate critical dissipative nonlinearity

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