Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T19:13:20.089Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON RESOLVENT CONVERGENCE ON A THIN DOMAIN

Part of: Partial differential equations

Published online by Cambridge University Press:  27 June 2013

RICARDO P. SILVA
RICARDO P. SILVA*
Affiliation:
Instituto de Geociências e Ciências Exatas, UNESP - Univ Estadual Paulista, Departamento de Matemática, 13506-900, Rio Claro SP, Brazil
*
[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we provide a new proof of strong convergence of resolvent operators associated with boundary value problems on thin domains.

Keywords

thin domainsresolvent convergence

MSC classification

Secondary: 35B25: Singular perturbations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 141 - 148
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Antoci, F. and Prizzi, M., ‘Reaction–diffusion equations on unbounded thin domains’, Topol. Methods Nonlinear Anal. 18 (2001), 283–302.Google Scholar
Arrieta, J. M., Bezerra, F. D. M. and Carvalho, A. N., ‘Rate of convergence of attractors for some singularly perturbed parabolic problems’, Topol. Methods Nonlinear Anal. (2013), to appear.Google Scholar
Arrieta, J. M. and Carvalho, A. N., ‘Spectral convergence and nonlinear dynamics of reaction–diffusion equations under perturbations of the domain’, J. Differential Equations 199 (2004), 143–178.Google Scholar
Arrieta, J. M., Carvalho, A. N. and Lozada-Cruz, G., ‘Dynamics in dumbbell domains I. Continuity of the set of equilibria’, J. Differential Equations 231 (2) (2006), 551–597.CrossRefGoogle Scholar
Arrieta, J. M., Carvalho, A. N. and Lozada-Cruz, G., ‘Dynamics in dumbbell domains II. The limiting problem’, J. Differential Equations 247 (2009), 174–202.Google Scholar
Arrieta, J. M., Carvalho, A. N. and Lozada-Cruz, G., ‘Dynamics in dumbbell domains III. Continuity of Attractors’, J. Differential Equations 231 (2009), 225–259.Google Scholar
Arrieta, J. M., Carvalho, A. N., Pereira, M. C. and Silva, R. P., ‘Semilinear parabolic problems in thin domains with a highly oscillatory boundary’, Nonlinear Anal. 74 (2011), 5111–5132.CrossRefGoogle Scholar
Carbone, V. L., Carvalho, A. N. and Schiabel-Silva, K., ‘Continuity of attractors for parabolic problems with localized large diffusion’, Nonlinear Anal. 68 (2008), 515–535.Google Scholar
Carvalho, A. N. and Piskarev, S., ‘A general approximation scheme for attractors of abstract parabolic problems’, Numer. Funct. Anal. Optim. 27 (7–8) (2006), 785–829.CrossRefGoogle Scholar
Ciuperca, I. S., ‘Reaction–diffusion equations on thin domains with varying order of thinness’, J. Differential Equations 126 (1996), 244–291.Google Scholar
Elsken, T., ‘Limiting behavior of attractors for systems on thin domains’, Hiroshima Math. J. 32 (2002), 389–415.CrossRefGoogle Scholar
Elsken, T., ‘Attractors for reaction–diffusion equations on thin domains whose linear part is nonself-adjoint’, J. Differential Equations 206 (2004), 94–126.Google Scholar
Elsken, T., ‘A reaction–diffusion equation on a net-shaped thin domain’, Studia Math. 165 (2004), 159–199.Google Scholar
Elsken, T., ‘Continuity of attractors for net-shaped thin domain’, Topol. Methods Nonlinear Anal. 26 (2005), 315–354.Google Scholar
Hale, J. K. and Raugel, G., ‘Reaction–diffusion equations on thin domains’, J. Math. Pures Appl. 71 (1) (1992), 33–95.Google Scholar
Prizzi, M., Rinaldi, M. and Rybakowski, K. P., ‘Curved thin domains and parabolic equations’, Studia Math. 151 (2002), 109–140.CrossRefGoogle Scholar
Prizzi, M. and Rybakowski, K. P., ‘The effect of domain squeezing upon the dynamics of reaction–diffusion equations’, J. Differential Equations 173 (2001), 271–320.Google Scholar
Raugel, G., Dynamics of Partial Differential Equations on Thin Domains, Lecture Notes in Mathematics, 1609 (Springer, Berlin, Heidelberg, 1995).CrossRefGoogle Scholar
Rekalo, A. M., ‘Asymptotic behavior of solutions of nonlinear parabolic equations on two-layer thin domains’, Nonlinear Anal. 52 (2003), 1393–1410.Google Scholar