[Paper Review] L^p-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

Yacine Chitour, Swann Marx+1 more|arXiv (Cornell University)|Jul 26, 2019

Stability and Controllability of Differential Equations26 references4 citations

TL;DR

This paper establishes well-posedness and exponential $L^p$-asymptotic stability for a 1D wave equation with nonlinear damping in $L^p$-based functional spaces for $p \in [2, \infty]$. Using a Lyapunov functional and an attractivity result for infinite-dimensional linear time-varying systems, it proves exponential decay of trajectories with explicit decay rate estimates, even for non-monotone nonlinearities, under Dirichlet boundary conditions and localized damping.

ABSTRACT

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping with an L p functional framework, p $\in$ [2, $\infty$]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results are proved by considering an appropriate functional of the energy in the desired functional spaces introduced by Haraux in [11]. Asymptotic behavior analysis is based on an attractivity result on a trajectory of an infinite-dimensional linear time-varying system with a special structure, which relies on the introduction of a suitable Lyapunov functional. Note that some of the results of this paper apply for a large class of nonmonotone dampings.

Motivation & Objective

• To establish well-posedness and asymptotic stability for a 1D wave equation with nonlinear damping in $L^p$-based functional spaces for $p \in [2, \infty]$.
• To analyze the asymptotic behavior of trajectories under general nonlinear damping, including non-monotone cases.
• To provide explicit decay rate estimates for the energy and state trajectories in $L^p$ norms.
• To extend stability results beyond the classical $L^2$ framework to $L^p$ spaces with $p \neq 2$, particularly for $p = \infty$.
• To address the challenge of non-boundedness of the D’Alembertian operator in $L^p$ for $p \neq 2$ in higher dimensions, focusing on the 1D case where boundedness holds.

Proposed method

• The analysis uses a functional energy framework introduced by Haraux in [11], adapted to $L^p$ spaces for $p \in [2, \infty]$.
• Well-posedness is established via energy estimates in the functional spaces $H_p(0,1) = W_0^{1,p}(0,1) \times L^p(0,1)$ and $D_p(0,1) = W^{2,p}(0,1) \cap W_0^{1,p}(0,1) \times W_0^{1,p}(0,1)$.
• A key technique involves analyzing a linear time-varying system derived from linearizing the original equation around a trajectory.
• The asymptotic stability is proven using a Lyapunov functional tailored to the time-varying system, leveraging an attractivity result for infinite-dimensional systems.
• Duhamel’s formula is applied to express the solution of the linearized system as a sum of homogeneous and inhomogeneous terms.
• Rellich-Kondrachov compactness embedding is used to upgrade $H^1$-boundedness to $L^\infty$-boundedness for the spatial derivative.

Experimental results

Research questions

• RQ1 Can exponential $L^p$-asymptotic stability be established for a 1D wave equation with nonlinear damping in $L^p$ spaces for $p \in [2, \infty]$?
• RQ2 How does the decay rate of trajectories depend on initial conditions and damping properties in these $L^p$ frameworks?
• RQ3 Can stability results be extended to non-monotone nonlinearities $\sigma$ satisfying $\xi \sigma(\xi) \geq 0$?
• RQ4 What is the role of the D’Alembertian operator in $L^p$ spaces for $p \neq 2$, and under what conditions is it bounded in 1D?
• RQ5 Can the stability analysis be extended to other PDEs such as the Korteweg-de Vries equation or boundary-controlled wave equations in $L^p$?

Key findings

• The system is well-posed in the functional spaces $H_p(0,1)$ and $D_p(0,1)$ for $p \in [2, \infty]$, with solutions existing and being unique under appropriate assumptions on $a(x)$ and $\sigma$.
• Exponential decay of trajectories to zero is proven in $H_p(0,1)$ with a decay rate depending on the initial condition size, for $p \in [2, \infty]$, under the assumption $\xi \sigma(\xi) \geq 0$.
• For initial data in $D_2(0,1)$, the solution satisfies $\|z_x\|_{L^\infty(0,1)} \leq C e^{-\beta t} \|(z_0,z_1)\|_{D_2(0,1)}$ for some $C, \beta > 0$, implying $L^\infty$-boundedness of the spatial derivative.
• The decay rate estimate in $H_\infty(0,1)$ is derived as $\|(z,z_t)\|_{H_\infty(0,1)} \leq C e^{-\beta_1 t} \|(z_0,z_1)\|_{D_2(0,1)}$, with $C = 2K_1 + \frac{K_2}{\beta_2 - \beta_1} m a_1$.
• The results apply to a broad class of non-monotone dampings, not restricted to sector-bounded or monotone nonlinearities.
• The analysis confirms that the D’Alembertian operator is bounded in $H_p(0,1)$ for $p \in [2, \infty]$ in one dimension, enabling the $L^p$ framework.