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Note on Boundary Stabilization of Degenerate Schrödinger Equations

Abdelkader Benaissa, Abbes Benaissa

TL;DR

This work addresses boundary stabilization for degenerate Schrödinger equations with a singular fractional-integral damping term, comparing damping at degenerate and nondegenerate boundaries. The authors reformulate the problems into a coupled PDE-ODE framework by introducing an auxiliary variable and establish a dissipative energy structure within a Hilbert space, enabling semigroup-based analysis. Through precise resolvent estimates near zero and an application of the Batty-Chill-Tomilov theorem, they prove polynomial energy decay rates for the associated semigroups, with explicit rates $E_1(t) \lesssim t^{-2}$ and $E_2(t) \lesssim t^{-\frac{2}{2-\beta}}$ (and $E_2(t) \lesssim t^{-2}$ when $\kappa(x)=x^{\alpha}$, $0<\alpha<2$). These results extend boundary stabilization theory to degenerate Schrödinger equations and provide quantitative decay rates for systems with degenerate boundaries and fractional damping.

Abstract

A degenerate Schrödinger equation under fractional integral damping is considered. Here the damping term is singular and not integrable and we consider the two cases when damping acting on the degenerate boundary and nondegenerate boundary. In this paper, we establish polynomial energy decay rates for the degenerate Schrödinger equation by using resolvent estimates.

Note on Boundary Stabilization of Degenerate Schrödinger Equations

TL;DR

This work addresses boundary stabilization for degenerate Schrödinger equations with a singular fractional-integral damping term, comparing damping at degenerate and nondegenerate boundaries. The authors reformulate the problems into a coupled PDE-ODE framework by introducing an auxiliary variable and establish a dissipative energy structure within a Hilbert space, enabling semigroup-based analysis. Through precise resolvent estimates near zero and an application of the Batty-Chill-Tomilov theorem, they prove polynomial energy decay rates for the associated semigroups, with explicit rates and (and when , ). These results extend boundary stabilization theory to degenerate Schrödinger equations and provide quantitative decay rates for systems with degenerate boundaries and fractional damping.

Abstract

A degenerate Schrödinger equation under fractional integral damping is considered. Here the damping term is singular and not integrable and we consider the two cases when damping acting on the degenerate boundary and nondegenerate boundary. In this paper, we establish polynomial energy decay rates for the degenerate Schrödinger equation by using resolvent estimates.
Paper Structure (3 sections, 3 theorems, 53 equations)

This paper contains 3 sections, 3 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result

Key Result

Theorem 2.1

Let $S(t)$ be a bounded $C_0$-semigroup on a Hilbert space ${\cal X}$ with generator ${\cal A}$. Assume that $\sigma({\cal A})\cap i{\bf \hbox{\sc I R}}=\{0\}$ and that there exist $\vartheta\geq 1$ and $\upsilon > 0$ such that Then there exist constants $C, t_0> 0$ such that for all $t\geq t_0$ and $U_0\in D({\cal A})\cap R({\cal A})$ we have

Theorems & Definitions (3)

  • Theorem 2.1: batty
  • Theorem 3.1
  • Theorem 3.2