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Finite time extinction for a critically damped Schr{ö}dinger equation with a sublinear nonlinearity

Pascal Bégout, Jesús Ildefonso Díaz

Abstract

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{ö}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^α}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2α=(1-m)$ and $m\in[0,1).$ Here we consider the sublinear case $0<m<1$ with a critical damped coefficient: $a\in\mathbb{C}$ is assumed to be in the set $D(m)=\big\{z\in\mathbb{C}; \; \mathrm{Im}(z)>0 \text{ and } 2\sqrt{m}\mathrm{Im}(z)=(1-m)\mathrm{Re}(z)\big\}.$ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropiate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.

Finite time extinction for a critically damped Schr{ö}dinger equation with a sublinear nonlinearity

Abstract

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{ö}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' with and Here we consider the sublinear case with a critical damped coefficient: is assumed to be in the set Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropiate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.
Paper Structure (7 sections, 19 theorems, 98 equations)

This paper contains 7 sections, 19 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of solutions
  3. Finite time extinction and asymptotic behavior
  4. Proofs of the existence of solutions
  5. On the $\boldsymbol{H^2}$-solutions
  6. Proofs of the finite time extinction and asymptotic behavior theorems
  7. Concluding remarks

Key Result

Proposition 2.5

Assume O, V and pV. Let $0<m\leqslant1,$$a\in\mathbb{C}$ and $f\in L^1_\mathrm{loc}([0,\infty);L^2(\Omega)).$ Let $u$ be a weak solution to nls. Let $(f_n,u_n)_{n\in\mathbb{N}}$ satisfy cv, where each $u_n$ is an $H^1_0$-solution to nls--nlsb with $f_n$ instead of $f.$ Then, and $u$ solves nls in $L^1_\mathrm{loc}([0,\infty);H^{-2}(\Omega)+L^\frac{2}{m}(\Omega))$ and so in $\mathscr{D}^\prime((0,

Theorems & Definitions (25)

  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6: Uniqueness and continuous dependance
  • Theorem 2.7: Existence and uniqueness of $\boldsymbol{L^2}$-solutions
  • Remark 2.8
  • Theorem 2.9: Additional regularity in $\boldsymbol{H^1_0}$ for weak solutions
  • Theorem 2.10: Existence and uniqueness of $\boldsymbol{H^1_0}$-solutions -- I
  • Theorem 2.11: Existence and uniqueness of $\boldsymbol{H^1_0}$-solutions -- II
  • ...and 15 more