[Paper Review] Linear stabilization for a degenerate wave equation in non divergence form with drift

Genni Fragnelli, Dimitri Mugnai|arXiv (Cornell University)|Dec 10, 2022

Stability and Controllability of Differential Equations4 citations

TL;DR

This paper establishes conditions for uniform exponential stabilization of a one-dimensional degenerate wave equation with drift, where the principal operator is in non-divergence form and degenerates at the boundary. By introducing a weighted energy functional and applying multiplier methods with Feller-type weights, the authors prove exponential decay of solutions under specific structural conditions on the degeneracy and drift coefficients.

ABSTRACT

We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Motivation & Objective

• To address the lack of stabilization results for degenerate wave equations in non-divergence form with drift.
• To extend existing stability theory beyond non-degenerate and divergence-form settings.
• To establish sufficient conditions for uniform exponential decay of energy in the presence of boundary damping and degeneracy at x=0.
• To develop a functional framework and energy estimates suitable for non-divergence form operators with singular coefficients.
• To generalize classical stabilization techniques to degenerate, non-divergence form PDEs using Feller-type weights and multiplier methods.

Proposed method

• Introduces a weighted energy functional involving the Feller weight $\eta(x) = \exp\left(\int_{1/2}^x \frac{b(s)}{a(s)}ds\right)$ to handle the non-divergence form structure.
• Applies multiplier methods adapted to degenerate operators by constructing a suitable test function and integrating by parts in weighted Sobolev spaces.
• Uses the condition $\frac{b}{a} \in L^1(0,1)$ to ensure integrability of the drift term in the energy estimates.
• Imposes structural assumptions on $a(x)$ and $b(x)$: weak or strong degeneracy with $K < 2$ in the degeneracy index $K = \sup \frac{x|a'(x)|}{a(x)}$.
• Employs integration by parts and Hölder’s inequality in weighted $L^2$ spaces to control boundary terms and prove vanishing limits at $x=0$.
• Establishes $W^{1,1}(0,1)$ regularity of key boundary terms to justify limit existence and prove vanishing at the degenerate boundary.

Experimental results

Research questions

• RQ1 Under what conditions does a degenerate wave equation in non-divergence form with drift exhibit uniform exponential energy decay?
• RQ2 How can boundary damping be effectively combined with degeneracy and drift to stabilize the system?
• RQ3 What role does the Feller-type weight $\eta(x)$ play in controlling the behavior of solutions near the degenerate boundary?
• RQ4 Can classical multiplier methods be adapted to non-divergence form degenerate PDEs with singular coefficients?
• RQ5 What structural assumptions on $a(x)$ and $b(x)$ ensure the necessary integrability and vanishing of boundary terms in energy estimates?

Key findings

• The paper proves that under the condition $\frac{b}{a} \in L^1(0,1)$ and appropriate degeneracy (weak or strong with $K < 2$), the energy of solutions decays uniformly exponentially.
• The boundary term at $x=1$ with damping $y_t + \eta y_x + \beta y = 0$ ensures energy dissipation, provided $\beta \geq 0$.
• The limit $\lim_{x \to 0} \eta(x) u(x) y'(x) = 0$ is rigorously established, which is essential for energy estimates.
• The vanishing of $\lim_{x \to 0} x \eta(x) (u'(x))^2 = 0$ is proven, ensuring integrability and regularity near the degenerate point.
• The condition $\frac{x}{a(x)} u^2(x) \to 0$ as $x \to 0$ is verified, which is necessary for the weighted energy to be finite.
• The proof relies on showing that certain boundary terms are in $W^{1,1}(0,1)$, hence their limits exist and are finite, enabling integration by parts.