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Upper semicontinuity of the attractor for a nonlinear hyperbolic-parabolic coupled system with fractional Laplacian

Manoel J. Dos Santos, Renato F. C. Lobato

Abstract

In this paper we establish the existence and uniqueness of global solutions (in time), as well as the existence, regularity and stability (upper semicontinuity) of the attractor for the semigroup generated by the solutions of a two-dimensional nonlinear hyperbolic-parabolic coupled system with fractional Laplacian. In addition, we also obtain the existence of an exponential attractor and show that this attractor has a finite fractal dimension in a space containing the phase space of the dynamical system.

Upper semicontinuity of the attractor for a nonlinear hyperbolic-parabolic coupled system with fractional Laplacian

Abstract

In this paper we establish the existence and uniqueness of global solutions (in time), as well as the existence, regularity and stability (upper semicontinuity) of the attractor for the semigroup generated by the solutions of a two-dimensional nonlinear hyperbolic-parabolic coupled system with fractional Laplacian. In addition, we also obtain the existence of an exponential attractor and show that this attractor has a finite fractal dimension in a space containing the phase space of the dynamical system.
Paper Structure (10 sections, 131 equations)

This paper contains 10 sections, 131 equations.

Table of Contents

  1. Introduction
  2. Notations and Assumptions
  3. Energy of Solutions
  4. Existence of Solution
  5. Global Attractor and Finite Dimensionality
  6. Quasi-Stability
  7. Gradient System
  8. Regularity of trajetories and exponential attractors
  9. Upper semicontinuity
  10. Known results for infinite-dimensional dynamical systems

Theorems & Definitions (13)

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  • proof : Proof of Theorem \ref{['thm:4:1']}
  • proof
  • ...and 3 more