Optimal L2-decay of solutions to a cubic dissipative nonlinear Schrödinger equation

Article type: Research Article

Authors: Naoyasu, Kita^{a; *} | Takuya, Sato^b

Affiliations: [a] Faculty of Advanced Science and Technology, Kumamoto University, Kurokami 2-39-1, Kumamoto, 860-8555, Japan. E-mail: [email protected] | [b] Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

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|2u. Our aim is to obtain the two results. One asserts that, if the L2-norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than (logt)−1/2, then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of (logt)−1/2 in L2.

Keywords: Nonlinear Schrödinger equation, decay estimate, critical dissipative nonlinearity

DOI: 10.3233/ASY-211738

Journal: Asymptotic Analysis, vol. 129, no. 3-4, pp. 505-517, 2022