Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

<jats:p>We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation <jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0308210500000561disp001" /></jats:disp-formula> where <jats:italic>L</jats:italic> is a linear pseudodifferential operator with dissipative symbol Re<jats:italic>L</jats:italic>(ξ) ≥ <jats:italic>C</jats:italic><jats:sub>1</jats:sub>|ξ|<jats:sup>2</jats:sup>/(1 + ξ<jats:sup>2</jats:sup>) and |<jats:italic>L</jats:italic>′(ξ)| ≤ <jats:italic>C</jats:italic><jats:sub>2</jats:sub>(|ξ|+ |ξ|<jats:sup><jats:italic>n</jats:italic></jats:sup>) for all ξ ∈ <jats:bold><jats:italic>R</jats:italic></jats:bold>. Here, <jats:italic>C</jats:italic><jats:sub>1</jats:sub>, <jats:italic>C</jats:italic><jats:sub>2</jats:sub> > 0, <jats:italic>n</jats:italic> ≥ 1. Moreover, we assume that <jats:italic>L</jats:italic>(ξ) = αξ<jats:sup>2</jats:sup> + <jats:italic>O</jats:italic>(|ξ|<jats:sup>2+γ</jats:sup>) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ<jats:sup>2</jats:sup>, equation (A) is the nonlinear Schrödinger equation with dissipation <jats:italic>u</jats:italic><jats:sub><jats:italic>t</jats:italic></jats:sub> − α<jats:italic>u<jats:sub>xx</jats:sub></jats:italic> + i|<jats:italic>u</jats:italic>|<jats:sup>2</jats:sup><jats:italic>u</jats:italic> = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate <jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0308210500000561disp002" /></jats:disp-formula> under the conditions that <jats:italic>u</jats:italic><jats:sub>0</jats:sub> ∈ <jats:bold><jats:italic>H</jats:italic></jats:bold><jats:sup><jats:italic>n</jats:italic>,0</jats:sup> ∩ <jats:bold><jats:italic>H</jats:italic></jats:bold><jats:sup>0,1</jats:sup> have the mean value <jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0308210500000561disp003" /></jats:disp-formula> and the norm ‖<jats:italic>u</jats:italic><jats:sub>0</jats:sub>‖<jats:sub><jats:bold><jats:italic>H</jats:italic></jats:bold><jats:sup><jats:italic>n</jats:italic>,0</jats:sup></jats:sub> + ‖<jats:italic>u</jats:italic><jats:sub>0</jats:sub>‖<jats:sub><jats:bold><jats:italic>H</jats:italic></jats:bold><jats:sup>0,1</jats:sup></jats:sub> = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and <jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0308210500000561disp004" /></jats:disp-formula> Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation <jats:italic>u</jats:italic><jats:sub><jats:italic>t</jats:italic></jats:sub> − α<jats:italic>u<jats:sub>xx</jats:sub></jats:italic> + i|<jats:italic>u</jats:italic>|<jats:sup><jats:italic>p</jats:italic>−1</jats:sup><jats:italic>u</jats:italic> = 0, with <jats:italic>p</jats:italic> > 3 have the same time decay estimate ‖<jats:italic>u</jats:italic>‖<jats:sub><jats:bold><jats:italic>L</jats:italic></jats:bold></jats:sub>∞ = <jats:italic>O</jats:italic>(<jats:italic>t</jats:italic><jats:sup>−½</jats:sup>) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.</jats:p>