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Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

Antonio Arnal

TL;DR

The paper analyzes the resolvent bounds for the generator G of the one-dimensional damped wave equation with unbounded damping a and nonnegative potential q. It reduces the problem to a λ-dependent quadratic operator T(λ)=H_q+2λ a+λ^2, and proves sharp asymptotics for ||T(λ)^{-1}|| in Fourier space via a turning-point (Airy) analysis, yielding ||T(λ)^{-1}|| ≈ ||(A-c)^{-1}||/(2|b|) (1+O(1/|b|)) for λ=-c+i b with c in a bounded set. These T(λ) estimates translate, through a Schur-complement argument, into resolvent bounds for G: ||(G-λ)^{-1}|| stays approximately constant on bounded-width vertical strips in the left half-plane as |Im λ|→∞. Under additional growth conditions on a and q, the generated semigroup is uniformly exponentially stable, and the paper provides a detailed example with a(x)=x^2, q(x)=κ x^2, describing the spectrum and decay rates. Overall, the work advances pseudospectral understanding of non-self-adjoint damped wave generators with unbounded damping by reducing the problem to a tractable Airy-type model and rigorously transferring bounds back to G.

Abstract

We study the generator $G$ of the one-dimensional damped wave equation with unbounded damping. We show that the norm of the corresponding resolvent operator, $\| (G - λ)^{-1} \|$, is approximately constant as $|λ| \to +\infty$ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, $\overline{\mathbb{C}}_{-} := \{λ\in \mathbb{C}: \operatorname{Re} λ\le 0\}$. Our proof rests on a precise asymptotic analysis of the norm of the inverse of $T(λ)$, the quadratic operator associated with $G$.

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

TL;DR

The paper analyzes the resolvent bounds for the generator G of the one-dimensional damped wave equation with unbounded damping a and nonnegative potential q. It reduces the problem to a λ-dependent quadratic operator T(λ)=H_q+2λ a+λ^2, and proves sharp asymptotics for ||T(λ)^{-1}|| in Fourier space via a turning-point (Airy) analysis, yielding ||T(λ)^{-1}|| ≈ ||(A-c)^{-1}||/(2|b|) (1+O(1/|b|)) for λ=-c+i b with c in a bounded set. These T(λ) estimates translate, through a Schur-complement argument, into resolvent bounds for G: ||(G-λ)^{-1}|| stays approximately constant on bounded-width vertical strips in the left half-plane as |Im λ|→∞. Under additional growth conditions on a and q, the generated semigroup is uniformly exponentially stable, and the paper provides a detailed example with a(x)=x^2, q(x)=κ x^2, describing the spectrum and decay rates. Overall, the work advances pseudospectral understanding of non-self-adjoint damped wave generators with unbounded damping by reducing the problem to a tractable Airy-type model and rigorously transferring bounds back to G.

Abstract

We study the generator of the one-dimensional damped wave equation with unbounded damping. We show that the norm of the corresponding resolvent operator, , is approximately constant as on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, . Our proof rests on a precise asymptotic analysis of the norm of the inverse of , the quadratic operator associated with .
Paper Structure (18 sections, 15 theorems, 246 equations, 2 figures)

This paper contains 18 sections, 15 theorems, 246 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Fourier transform and pseudo-differential operators
  4. Schrödinger operators with complex potentials
  5. Generalised Airy operators
  6. The damped wave equation
  7. Assumptions and statement of the main result
  8. The associated quadratic operator
  9. Proof of Theorem \ref{['thm:resolvent.Tla']}
  10. Step 1: estimate outside the neighbourhoods of $\pm \xi_b$
  11. Step 2: estimate near $\pm \xi_b$
  12. Step 3: lower estimate
  13. Step 4: combining the estimates
  14. The norm of the resolvent along curves adjacent to the imaginary axis
  15. The operator $G$
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $F \in {C^{\infty}({\mathbb{R}})}$ and $m > 0$ be such that and let $\phi \in {C^{\infty}({\mathbb{R}})} \cap {L^{\infty}({\mathbb{R}})}$ be such that $\operatorname{supp} \phi'$ is bounded. For $j \in \mathbb{N}_0$ and $u \in {\mathscr S({\mathbb{R}})}$, we define the operators (with $P:=P^{(0)}$ and $Q:=Q^{(0)}$) Then, for any $N \in \mathbb{N}_0$, we have where $R_{N+1}$ is a pseudo-diff

Figures (2)

  • Figure 6.1: The spectrum of $G$ for $q(x) = 10 x^2$ and $a(x) = x^2$.
  • Figure 6.2: First 5 real (top) and first 5 imaginary (bottom) eigenvalues of $G$ as a function of $\kappa$.

Theorems & Definitions (37)

  • Lemma 2.1
  • Example 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • Remark 3.7
  • Lemma 3.8
  • proof
  • Corollary 3.9
  • ...and 27 more