Parabolic equations with localized large diffusion: Rate of convergence of attractors

Alexandre N. Carvalho, Leonardo Pires



In this paper we study the asymptotic nonlinear dynamics of scalar semilinear parabolic problems of reaction-diffusion type when the diffusion coefficient becomes large in a subregion in the interior to the domain. We obtain, under suitable assumptions, that the family of attractors behaves continuously and we exhibit the rate of convergence. An accurate description of the localized large diffusion is necessary.


Localized large diffusion; reaction diffusion equations; rate of convergence; attractors

Full Text:



J. M. Arrieta, F.D. Bezerra and A.N. Carvalho, Rate of convergence of global attractors of some perturbed reaction-diffusion problems, Topol. Methdos Nonlinear Anal. 41 (2013), no. 2, 229–253.

J.M. Arrieta, A.N. Carvalho and A. Rodrı́guez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations 168 (2000), no. 1, 33–59.

J.M. Arrieta, A.N. Carvalho and A. Rodrı́guez-Bernal Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), no. 2, 376–406.

J.M. Arrieta, A.N. Carvalho and A. Rodrı́guez-Bernal, Attractors for parabolic problems with nonlinear boundary bondition. Uniform bounds, Commun. Partial Differential Equations 25 (2000), no. 1–2, 1–37.

J.M. Arrieta and E. Santamaria, Distance of attractors of reaction-diffusion equations in thin domains, J. Differential Equations 263 (2017), no. 9, 5459–5506.

M.C. Bortolan, A.N. Carvalho, J.A. Langa and G. Raugel, Non-autonomous perturbations of morse-smale semigroups: stability of the phase diagram, Preprint.

V.L. Carbone, A.N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Anal. 68 (2008), no. 3, 515–535.

V.L. Carbone and J.G. Ruas-Filho, Continuity of the attractors in a singular problem arising in composite materials, Nonlinear Anal. 65 (2006), 1285–1305.

A.N. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Springer, 2010.

A.N. Carvalho and A.L. Pereira, A sacalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations, J. Differential Equations 112 (1994), 81–130.

A.N. Carvalho and L. Pires, Rate of convergence of attractors for singularly perturbed semilinear problems, J. Math. Anal. Appl. 452 (2017), no. 1, 258–296.

A.N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim. 27 (2006), 785–829.

G. Fusco, On the explicit construction of an ode which has the same dynamics as a scalar parabolic PDE, J. Differential Equations 69 (1987), 85–110.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer–Velag, 1980.

S.Y. Pilyugun, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, vol. 1706, Springer, 1999.

A. Rodrguez-Bernal, Localized spatial homogenizations and large diffusion, SIAM J. Math. Anal. 29 (1998), no. 6, 1361–1380.


  • There are currently no refbacks.

Partnerzy platformy czasopism