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Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks

Elaine A. Tavares-Lima, Bianca Lorenzi, Marcone C. Pereira

Abstract

In this work, we analyze the asymptotic behavior of the attractors associated with a semilinear parabolic equation subject to homogeneous Neumann boundary conditions and defined on a thin domain $R^\varepsilon \subset \mathbb{R}^{1+n}$. We assume that the thin domain exhibits a cusp, known as an outward peak, whose geometry is characterized by a nonnegative function that vanishes at a point on the boundary. Our objective is to rigorously establish the continuity of the attractors as $\varepsilon \to 0$ and to determine their rate of convergence.

Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks

Abstract

In this work, we analyze the asymptotic behavior of the attractors associated with a semilinear parabolic equation subject to homogeneous Neumann boundary conditions and defined on a thin domain . We assume that the thin domain exhibits a cusp, known as an outward peak, whose geometry is characterized by a nonnegative function that vanishes at a point on the boundary. Our objective is to rigorously establish the continuity of the attractors as and to determine their rate of convergence.
Paper Structure (16 sections, 30 theorems, 224 equations, 1 figure)

This paper contains 16 sections, 30 theorems, 224 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary settings
  3. Domain Construction
  4. Parabolic Problem
  5. Well-posedness and existence of a global attractor
  6. Estimates for the Elliptic Problem
  7. Terms converging to zero:
  8. Compacity of the resolvent of the limit operator
  9. Abstract settings and analysis of the nonlinear terms
  10. Rate of the distance of attractors
  11. Continuity of the equilibria set
  12. Convergence of linear semigroups
  13. Continuity of nonlinear semigroups
  14. Linearization
  15. Rate of convergence and attraction of local unstable manifolds
  16. ...and 1 more sections

Key Result

Lemma 2.1

If a satisfies the condition H1 and $u \in H^1_\varepsilon(\Omega)$, then $M_\varepsilon u \in H^1_a(0,1)$. Moreover, the derivative of $M_\varepsilon u$ satisfies the identity

Figures (1)

  • Figure 1: Comparison of thin domains for different profiles $a(x)$.

Theorems & Definitions (62)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3: Local well-posedness
  • Theorem 3.4: Boundedness of global solutions
  • Theorem 3.5: Global attractor
  • ...and 52 more