A frequency approach for stabilization of one-dimensional degenerate wave equation

TL;DR

This paper investigates the stabilization of a strongly degenerate one-dimensional wave equation using a frequency domain approach, demonstrating polynomial energy decay at a rate of t^{-1/2}.

Contribution

It introduces a frequency domain method to establish polynomial decay rates for the energy of a degenerate wave equation, extending previous approaches to this class of problems.

Findings

Proves polynomial decay of energy at rate t^{-1/2}

Uses frequency domain method inspired by prior work

Addresses stabilization for degenerate wave equations

Abstract

In this paper, we are concerned with the study of stabilization problem for the following strongly degenerate wave equation in one space dimension $$w_{tt}(x,t)-\left(x^\alpha w_x(x,t)\right)_x=0$$ where ${\bf\alpha\in [1,2)}$. Thus, using a frequency domain method inspired from \cite{BT}, we prove the polynomial decays of its total energy with $t^{-\nicefrac{1}{2}}$ decay rate.