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Boundary Exponential Stabilization for the Linear KP-II equation without Critical Size Restrictions

F. A. Gallego, J. R. Muñoz

TL;DR

This work addresses boundary stabilization for the linear KP-II equation on a rectangle $(0,L)\times(0,L)$ by designing a boundary feedback with $|\alpha|<1$ and $\beta>0$ that yields exponential decay of the energy $E(t)$ for all $L>0$. It provides a rigorous semigroup well-posedness framework, energy dissipation, and smoothing estimates, and then uses the Lions compactness-uniqueness method to derive an observability inequality, which implies global exponential stabilization. The main contribution is achieving exponential stabilization without critical-length restrictions, contrasting with prior works that exhibit length-dependent constraints. The results are built upon an anisotropic Sobolev framework and an anisotropic Gagliardo–Nirenberg inequality, and the nonlinear extension remains an open challenge with potential via damping/anti-damping or time-delay strategies.

Abstract

In this paper, we delve into the intricacies of boundary stabilization for the linearized KP-II equation within the constraints of a bounded domain, a phenomenon known as ``critical length." Our primary aim is to design a feedback law that ensures the existence and exponential stabilization of solutions in the energy space, without length restrictions on the domain $ Ω= (0, L) \times (0, L)$, $ L > 0 $. Furthermore, we examine the interaction between the drift term $ u_x $ under these constraints.

Boundary Exponential Stabilization for the Linear KP-II equation without Critical Size Restrictions

TL;DR

This work addresses boundary stabilization for the linear KP-II equation on a rectangle by designing a boundary feedback with and that yields exponential decay of the energy for all . It provides a rigorous semigroup well-posedness framework, energy dissipation, and smoothing estimates, and then uses the Lions compactness-uniqueness method to derive an observability inequality, which implies global exponential stabilization. The main contribution is achieving exponential stabilization without critical-length restrictions, contrasting with prior works that exhibit length-dependent constraints. The results are built upon an anisotropic Sobolev framework and an anisotropic Gagliardo–Nirenberg inequality, and the nonlinear extension remains an open challenge with potential via damping/anti-damping or time-delay strategies.

Abstract

In this paper, we delve into the intricacies of boundary stabilization for the linearized KP-II equation within the constraints of a bounded domain, a phenomenon known as ``critical length." Our primary aim is to design a feedback law that ensures the existence and exponential stabilization of solutions in the energy space, without length restrictions on the domain , . Furthermore, we examine the interaction between the drift term under these constraints.
Paper Structure (7 sections, 10 theorems, 72 equations)

This paper contains 7 sections, 10 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. State of the Art
  4. Notations and Main Result
  5. Well-posedness and Main Estimates
  6. Exponential Stabilization - Compactness Uniqueness approach
  7. Futher Comments

Key Result

Theorem 1.1

Let $\beta$ and $\alpha^{(j)}$, for $j = 1,\dots, N$, denote $n$- dimensional multi-indices with non-negative-integer-valued components. Suppose that $1 < p^{(j)} < \infty$, $1 < q < \infty$, $0 < \mu_j < 1$ with Then, for $f(x)\in C_0^\infty(\mathbb{R}^n)$, $\left\lVert D^\beta f \right\rVert_{q} \leq C \prod_{j=1}^N \left\lVert D^{\alpha^{(j)}}f \right\rVert_{p^{(j)}}^{\mu_j},$ where for non-

Theorems & Definitions (22)

  • Theorem 1.1: besov79
  • Theorem 1.2: Global Uniform Stabilization
  • Lemma 2.1
  • Remark 1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • ...and 12 more