One dimensional wave equation with in-domain localized damping and Wentzell boundary conditions

TL;DR

The paper analyzes exponential stability for a one-dimensional wave equation with in-domain localized damping and Wentzell boundary conditions. It develops two complementary approaches in the $L^2$ setting—a multiplier-based energy method and a Huang–Prüss-type frequency-domain analysis—to establish exponential decay toward an attractor value $u_*=u_0(1)-\eta_{2,0}$. It then extends the decay results to $L^p$ settings with $p>2$ under the additional assumption $a\equiv1$, using a combination of a D'Alembert-based well-posedness analysis and a Riemann-invariants formulation. Overall, the work provides a rigorous framework for exponential stabilization under localized interior damping and dynamic boundary conditions, with implications for vibration suppression in systems described by 1D wave dynamics.

Abstract

This paper is devoted to the exponential stability for one-dimensional linear wave equations with in-domain localized damping and several types of Wentzell (or dynamic) boundary conditions. In a quite general boundary setting, we establish the exponential decay of solutions towards the corresponding steady states. The results are obtained either by the multiplier method or spectral analysis in an $L^2$-functional framework, and then with input-to-state technics in an $L^p$-functional framework for $p \in (2,\infty)$.

Figures (1)

• Figure 1: Illustration of the functions $\phi$, $\psi$, and $\varphi$.

Theorems & Definitions (38)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof