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Generalized exponential pullback attractor for a nonautonomous wave equation

Matheus C. Bortolan, Tomas Caraballo, Carlos Pecorari Neto

Abstract

In this work we introduce the concept of generalized exponential $\mathfrak{D}$-pullback attractor for evolution processes, where $\mathfrak{D}$ is a universe of families in $X$, which is a compact and positively invariant family that pullback attracts all elements of $\mathfrak{D}$ with an exponential rate. Such concept was introduced in arXiv:2311.15630 for the general case of decaying functions (which include the exponential decay), but for fixed bounded sets rather than to universe of families. We prove a result that ensures the existence of a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for an evolution process, where $\mathfrak{D}_{\mathcal{C}^\ast}$ is a specific universe. This required an adaptation of the results of arXiv:2311.15630, which only covered the case of a polynomial rate of attraction, for fixed bounded sets. Later, we prove that a nonautonomous wave equation has a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor. This, in turn, also implies the existence of the $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for such problem.

Generalized exponential pullback attractor for a nonautonomous wave equation

Abstract

In this work we introduce the concept of generalized exponential -pullback attractor for evolution processes, where is a universe of families in , which is a compact and positively invariant family that pullback attracts all elements of with an exponential rate. Such concept was introduced in arXiv:2311.15630 for the general case of decaying functions (which include the exponential decay), but for fixed bounded sets rather than to universe of families. We prove a result that ensures the existence of a generalized exponential -pullback attractor for an evolution process, where is a specific universe. This required an adaptation of the results of arXiv:2311.15630, which only covered the case of a polynomial rate of attraction, for fixed bounded sets. Later, we prove that a nonautonomous wave equation has a generalized exponential -pullback attractor. This, in turn, also implies the existence of the -pullback attractor for such problem.
Paper Structure (7 sections, 19 theorems, 138 equations)

This paper contains 7 sections, 19 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Generalized exponential D-pullback attractors
  3. Existence of generalized exponential DCast-pullback attractors
  4. Application to a nonautonomous wave equation
  5. Auxiliary estimates
  6. Properties of uniform pullback absorption
  7. Auxiliary results

Key Result

Theorem 1.2

Assume that cond1-cond6 hold true. Then the evolution process $S$ associated with ourproblem in $X:=H^1_0(\Omega)\times L^2(\Omega)$ possesses a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor $\hat{M}\in \mathfrak{D}_{\mathcal{C}^\ast}$ in $X$. Furthermore, $S$ has a $\

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Proposition 3.1
  • ...and 24 more