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Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data. (English) Zbl 1167.35043
Summary: We study the asymptotic behavior in time of solutions to the Cauchy problem of nonlinear Schrödinger equations with a long-range dissipative nonlinearity given by $\lambda |u|^{p-1} u$ in one space dimension, where $1<p \le 3$ (namely, $p$ is a critical or subcritical exponent) and $\lambda$ is a complex constant satisfying Im $\lambda <0$ and $\left( (p-1)/2\sqrt{p} \right) |$Re $ \lambda | \le | $Im $\lambda |$ . We present the time decay estimates and the large-time asymptotics of the solution for arbitrarily large initial data, when $p=3$ or $p<3$ and $p$ is suitably close to 3.

35Q55NLS-like (nonlinear Schrödinger) equations
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI
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