Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T03:10:31.205Z Has data issue: false hasContentIssue false

CONVERGENCE OF SOLUTIONS TO SOME ELLIPTIC EQUATIONS IN BOUNDED NEUMANN THIN DOMAINS

Published online by Cambridge University Press:  20 November 2015

CÉSAR R. DE OLIVEIRA*
Affiliation:
Departamento de Matemática, UFSCar, São Carlos, SP 13560-970, Brazil email [email protected]
ALESSANDRA A. VERRI
Affiliation:
Departamento de Matemática, UFSCar, São Carlos, SP 13560-970, Brazil email [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.

Consider the (elliptic) stationary nonlinear reaction–diffusion equation in a sequence of bounded Neumann tubes in a space that is squeezed to a reference curve. It is supposed that the forcing term is square integrable and that the nonlinear one satisfies some growth and dissipative conditions. A norm convergence of the resolvents of the operators associated with the linear terms of such equations is proven, and this fact is used to provide new and simpler proofs of the asymptotic behaviour of the solutions to the full nonlinear equations (previously known in similar singular problems).

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

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), 143178.Google Scholar
Bouchitté, G., Mascarenhas, M. L. and Trabucho, L., ‘On the curvatures and torsion effects in one-dimensional waveguides’, ESAIM Control Optim. Calc. Var. 13 (2007), 793808.Google Scholar
Bouchitté, G., Mascarenhas, M. L. and Trabucho, L., ‘Thin waveguides with Robin boundary conditions’, J. Math. Phys. 53 (2012), 123517, 24 pages.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), 785829.Google Scholar
Elsken, T., ‘Limiting behavior of attractors for systems on thin domains’, Hiroshima Math. J. 32 (2002), 389415.CrossRefGoogle Scholar
Elsken, T., ‘A reaction–diffusion equation on a net-shaped thin domain’, Studia Math. 165 (2004), 159199.Google Scholar
Elsken, T., ‘Continuity of attractors for net-shaped thin domain’, Topol. Methods Nonlinear Anal. 26 (2005), 315354.CrossRefGoogle Scholar
Friedlander, L. and Solomyak, M., ‘On the spectrum of the Dirichlet Laplacian in a narrow infinite strip’, in: Spectral Theory of Differential Operators, American Mathematical Society Translation Series 2, 225 (American Mathematical Society, Providence, RI, 2008), 103116.Google Scholar
Hale, K. and Raugel, G., ‘Reaction-diffusion equation on thin domains’, J. Math. Pures Appl. 71 (1992), 3395.Google Scholar
Klingenberg, W., A Course in Differential Geometry (Springer, New York, 1978).CrossRefGoogle Scholar
de Oliveira, C. R., ‘Quantum singular operator limits of thin Dirichlet tubes via Γ-convergence’, Rep. Math. Phys. 67 (2011), 132.Google Scholar
de Oliveira, C. R. and Rossini, A. F., ‘Effective operators for Robin Laplacian in thin two- and three-dimensional curved waveguides’, Preprint.Google Scholar
de Oliveira, C. R. and Verri, A. A., ‘On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes’, J. Math. Anal. Appl. 381 (2011), 454468.CrossRefGoogle Scholar
de Oliveira, C. R. and Verri, A. A., ‘Norm resolvent convergence of Dirichlet Laplacian in unbounded thin waveguides’, Bull. Braz. Math. Soc. (N.S.) 46 (2015), 139158.Google Scholar
Prizzi, M., ‘A remark on reaction–diffusion equations in unbounded domains’, Discrete Contin. Dyn. Syst. 9 (2003), 281286.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), 271320.CrossRefGoogle Scholar
Rekalo, A. M., ‘Asymptotic behavior of solutions of nonlinear parabolic equations on two-layer thin domains’, Nonlinear Anal. 52 (2003), 13931410.Google Scholar
Silva, R. P., ‘A note on resolvent convergence on a thin domain’, Bull. Aust. Math. Soc. 89 (2014), 141148.Google Scholar