Topological Methods in Nonlinear Analysis

Rate of convergence of global attractors of some perturbed reaction-diffusion problems

José M. Arrieta, Flank D.M. Bezerra, and Alexandre N. Carvalho

Full-text: Access by subscription

Abstract

In this paper we treat the problem of the rate of convergence of attractors of dynamical systems for some autonomous semilinear parabolic problems. We consider a prototype problem, where the diffusion $a_0(\cdot)$ of a reaction-diffusion equation in a bounded domain $\Omega$ is perturbed to $a_\varepsilon(\cdot)$. We show that the equilibria and the local unstable manifolds of the perturbed problem are at a distance given by the order of $\|a_\varepsilon-a_0\|_\infty$. Moreover, the perturbed nonlinear semigroups are at a distance $\|a_\varepsilon-a_0\|_\infty^\theta$ with $\theta< 1$ but arbitrarily close to $1$. Nevertheless, we can only prove that the distance of attractors is of order $\|a_\varepsilon-a_0\|_\infty^\beta$ for some $\beta< 1$, which depends on some other parameters of the problem and may be significantly smaller than $1$. We also show how this technique can be applied to other more complicated problems.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 41, Number 2 (2013), 229-253.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461245476

Mathematical Reviews number (MathSciNet)
MR3114306

Zentralblatt MATH identifier
1331.35053

Citation

Arrieta, José M.; Bezerra, Flank D.M.; Carvalho, Alexandre N. Rate of convergence of global attractors of some perturbed reaction-diffusion problems. Topol. Methods Nonlinear Anal. 41 (2013), no. 2, 229--253. https://projecteuclid.org/euclid.tmna/1461245476


Export citation

References

  • J.M. Arrieta and A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations , Trans. Amer. Math. Soc., 352(2000), 285–310 \ref\key 2 ––––, Spectral convergence and nonlinear dynamics of reaction–diffusion equations under perturbations of the domain , J. Differential Equations, 199(2004), 143–178 \ref\key 3
  • J.M. Arrieta, A.N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains \romI: Continuity of the set of equilibria, J. Differential Equations, 231(2006), 551–597 \ref\key 4 ––––, Dynamics in dumbbell domains \romII: The limiting problem, J. Differential Equations, 247(2009), 174–202 \ref\key 5 ––––, Dynamics in dumbbell domains \romIII: Continuity of attractors, J. Differential Equations, 247(2009), 225–259 \ref\key 6
  • J.M. Arrieta, A.N. Carvalho and A. Rodriguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities , J. Differential Equations, 156(1999), 376–406 \ref\key 7 ––––, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds , Comm. Partial Differential Equations, 25(2000), 1–37 \ref\key 8 ––––, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions , J. Differential Equations, 168(2000), 33–59 \ref\key 9
  • J.M. Arrieta and E. Santamaria, Distance of attractors for Morse–Smale systems (2011). in preparation \ref\key 10
  • A.V. Babin and M.I. Vishik, Attractors of evolution equations , Stud. Math. Appl., 25 , North-Holland (1992) \ref\key 11 ––––, Regular attractors of semigroups and evolution equations , J. Math. Pures Appl., 62(1983), 441–491 \ref\key 12
  • S.M. Bruschi, A.N. Carvalho, J.W. Cholewa and T. Dlotko, Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations , J. Dynam. Differential Equations, 18(2006), 767–814 \ref\key 13
  • V.L. Carbone, A.N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion , Nonlinear Anal., 68(2008), 515–535 \ref\key 14
  • A.N. Carvalho and J.W. Cholewa, Exponential global attractors for semigroups in metric spaces with applications to differential equations , Ergodic Theory Dynam. Systems, to appear \ref\key 15
  • A.N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn–Hilliard equation , J. Math. Anal. Appl., 344(2008), 703–325 \ref\key 16
  • A.N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems , Numer. Funct. Anal. Optim., 27(2006), 785–829 \ref\key 17
  • J.K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications , Ann. Mat. Pura Appl., 154(1989), 281–326 \ref\key 18
  • D.B. Henry, Geometric Theory of Semilinear Parabolic Equations , Lecture Notes in Mathematics, 840 , Springer–Verlag (1981) \ref\key 19
  • T. Kato, Perturbation Theory of Linear Operators, Springer–Verlag (1976) \ref\key 20
  • A. Rodriguez-Bernal, Localized spatial homogenization and large diffusion , SIAM J. Math. Anal., 29(1998), 1361–1380