Stability of KdV equation on a network with bounded and unbounded branches

¹ Université Paul Sabatier, Institut de Mathématiques de Toulouse, France
² Université Grenoble Alpes, Laboratoire Jean Kuntzmann, France
³ Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, France

^* Corresponding author: hugo.parada.mat@gmail.com

Received: 19 January 2024
Accepted: 12 September 2024

Abstract

In this work, we studied the exponential stability of the nonlinear KdV equation posed on a star shaped network with a finite number of branches. On each branch of the network we define a KdV equation posed on a finite domain (0, ℓ_j) or the half-line (0,∞). We start by proving well-posedness and some regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on some branches. The main idea is to prove a suitable observability inequality. In the nonlinear case, we obtain two kinds of results: The first result holds for small amplitude solutions, and is proved using a perturbation argument from the linear case but without acting on all edges. The second result is a semiglobal stability result, and it is obtained by proving an observability inequality directly for the nonlinear system, but we need to act with damping terms on all the branches. In this case, we are able to prove the stabilization in weighted spaces.