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

Abstract

We study the Cauchy problem for the nonlinear Schr odinger equation with dissipation ut + L u + ijuj 2 u = 0;x 2 R; t > 0;where L is a linear pseudodi® erential operator with dissipative symbol Re L(¹ ) > C1 j¹ j 2 =(1 + ¹ 2 ) and jL 0 (¹ )j 6 C2 (j¹ j + j¹ j n ) for all ¹ 2 R. Here, C1 ; C2 > 0, n > 1. Moreover, we assume that L(¹ ) = ¬ ¹ 2 + O(j¹ j 2+ ® ) for all j¹ j < 1, where ® > 0, Re ¬ > 0, Im ¬ > 0. When L(¹ ) = ¬ ¹ 2 , equation (A) is the nonlinear Schr odinger equation with dissipation ut ¬ uxx + ijuj 2 u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimateunder the conditions that u0 2 H n;0 \ H 0;1 have the mean valuêand the norm ku0 k H n;0 + ku0 k H 0;1 = " is su± ciently small, where ¼ = 1 if Im ¬ > 0 and ¼ = 2 if Im ¬ = 0, and H m ;s = f¿ 2 S 0 ; k¿ km ;s = k(1 + x 2 ) s=2 (1 @ 2 x ) m =2 ¿ k < 1 g; m; s 2 R: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 ut ¬ uxx + ijuj p 1 u = 0, with p > 3 have the same time decay estimate kuk L 1 = O(t 1=2 ) 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.

Asymptotics for critical nonconvective type equations

“…In the super critical case, large-time asymptotics of solutions is similar to that for the linearized case (see [2,4,18,20] and the references cited therein). Critical and subcritical dissipative equations with nonconvective type nonlinearities were considered in papers [5,6,7,8,9,10,11,12,13,14,15,16,17,19,22].…”

“…In [12], we obtained the large-time asymptotic behavior of solutions to the Cauchy problem for the nonlinear Schrödinger equation with dissipation…”

“…Note that the method of [7,9] does not work for (1.1). In the present paper, we generalize the approach developed in our previous papers [10,12,13,14,16,17], where we considered mainly the case of Laplacian ᏸ = −∆, the power nonlinearity ᏺ(u) = |u| q 1 u,…”

Large Time Behavior of Small Solutions to Dirichlet Problem for Landau-Ginzburg Type Equations

“…[5], [6], [9], [10], [21] and references cited therein). Blow-up in finite time of positive solutions to the Cauchy problem for the heat equation u t À Du ¼ u 1þs was proved in [3], [8], [15], [22].…”

“…In the same way as in the proof of (16) We make a change of the dependent variable u ¼ ve ÀjðtÞþicðtÞ as in [9], where jðtÞ and cðtÞ are real valued functions and defined later. Then for the new function v we get the equation…”