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Finite time extinction for a class of damped Schr{ö}dinger equations with a singular saturated nonlinearity

Pascal Bégout, Jesús Ildefonso Díaz

Abstract

We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schr{ö}dinger equations 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 $m\in[0,1).$ To carry out the improvement of previous results in the literature we present in this paper a careful revision of the existence and regularity of weak solutions under very general assumptions on the data. We prove that the problem can be solved in the very general framework of the maximal monotone operators theory, even under a lack of regularity of the damping term. This allows us to consider, among other things, the singular case $m=0.$ We replace the above approximation of the damping term by a different one which keeps the monotonicity for any $\varepsilon\geqslant0$. We prove that, when $m=0,$ the finite extinction time of the solution arises for merely bounded right hand side data $f(t,x).$ This is specially useful in the applications in which the Schr{ö}dinger equation is coupled with some other functions satisfying some additional equations.

Finite time extinction for a class of damped Schr{ö}dinger equations with a singular saturated nonlinearity

Abstract

We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schr{ö}dinger equations when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' with and To carry out the improvement of previous results in the literature we present in this paper a careful revision of the existence and regularity of weak solutions under very general assumptions on the data. We prove that the problem can be solved in the very general framework of the maximal monotone operators theory, even under a lack of regularity of the damping term. This allows us to consider, among other things, the singular case We replace the above approximation of the damping term by a different one which keeps the monotonicity for any . We prove that, when the finite extinction time of the solution arises for merely bounded right hand side data This is specially useful in the applications in which the Schr{ö}dinger equation is coupled with some other functions satisfying some additional equations.
Paper Structure (7 sections, 29 theorems, 124 equations)

This paper contains 7 sections, 29 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Finite time extinction and asymptotic behavior
  4. On the zero-order terms
  5. Some maximal monotone operators
  6. Proofs of the existence theorems
  7. Proofs of the finite time extinction and asymptotic behavior theorems

Key Result

Proposition 2.4

Let Assumption ass1 be fulfilled and let $f\in L^1_\mathrm{loc}([0,\infty);L^2(\Omega)).$ If $u$ is a weak solution to nls then In addition, $u$ solves nls in $L^1_\mathrm{loc}([0,\infty);H^{-2}(\Omega)+L^\frac{2}{m}(\Omega))$ and so in $\mathscr{D}^\prime((0,\infty)\times\Omega).$

Theorems & Definitions (40)

  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5: Uniqueness and continuous dependance
  • Theorem 2.6: Existence and uniqueness of $\boldsymbol{L^2}$-solutions
  • Theorem 2.7: Additional regularity in $\boldsymbol{H^1_0}$
  • Remark 2.8
  • Theorem 2.9: Existence and uniqueness of $\boldsymbol{H^1_0}$-solutions
  • Theorem 2.10: Existence and uniqueness of $\boldsymbol{H^2}$-solutions
  • Remark 2.11
  • ...and 30 more