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Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations

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  • We study the upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations

    $\begin{matrix} i{{\partial }_{t}}u+\frac{1}{2}\Delta u=\lambda {{\left| u \right|}^{p-1}}u,\left( t,x \right)\in {{\mathbb{R}}^{+}}\text{ }\!\!\times\!\!\text{ }{{\mathbb{R}}^{n}}, \\ u\left( 0,x \right)={{u}_{0}}\left( x \right),x\in {{\mathbb{R}}^{n}}, \\ \end{matrix}$

    in space dimensions $n=1,2$ or $3$ , where $\lambda =\lambda _{1}+i\lambda _{2},$ $\lambda _{j}∈ \mathbb{R},$ $j=1,2,$ $\lambda _{2}<0$ and the subcritical order of nonlinearity $p=1+\frac{2}{n}-μ ,$ where $μ >0$ is small enough.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.


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