A reaction-diffusion equation on a family of three dimensional thin domains, collapsing onto a two dimensional subspace, is considered. In [The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations 173 (2001), 271–320] it was proved that, as the thickness of the domains tends to zero, the solutions of the equations converge in a strong sense to the solutions of an abstract semilinear parabolic equation living in a closed subspace of $H^1$. Also, existence and upper semicontinuity of the attractors was proved. In this work, for a specific class of domains, the limit problem is completely characterized as a system of two-dimensional reaction-diffusion equations, coupled by mean of compatibility and balance boundary conditions.
References
\ref \no \dfa arri1..J. Arrieta, Neumann eigenvalue problems on exterior perturbations of the domain , J. Differential Equations, 118 (1995), 54–103 MR1329403 10.1006/jdeq.1995.1067\ref \no \dfa arri1..J. Arrieta, Neumann eigenvalue problems on exterior perturbations of the domain , J. Differential Equations, 118 (1995), 54–103 MR1329403 10.1006/jdeq.1995.1067
\ref \no \dfa arriha..J. Arrieta, J. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains , J. Differential Equations, 91 (1991), 24–52 MR1106116 10.1016/0022-0396(91)90130-2\ref \no \dfa arriha..J. Arrieta, J. Hale and Q. Han, Eigenvalue problems for nonsmoothly perturbed domains , J. Differential Equations, 91 (1991), 24–52 MR1106116 10.1016/0022-0396(91)90130-2
\ref \no \dfa carry.. M. C. Carbinatto and K. P. Rybakowski, Conley index continuation and thin domain problems , Topol. Methods Nonlinear Anal., 16 (2000), 201–251 MR1820507\ref \no \dfa carry.. M. C. Carbinatto and K. P. Rybakowski, Conley index continuation and thin domain problems , Topol. Methods Nonlinear Anal., 16 (2000), 201–251 MR1820507
\ref \no \dfa Ciu1..I. S. Ciuperca, Spectral properties of Schrödinger operators on domains with varying order of thinness , J. Dynam. Differential Equations, 10 (1998), 73–108 MR1607535 10.1023/A:1022640429041\ref \no \dfa Ciu1..I. S. Ciuperca, Spectral properties of Schrödinger operators on domains with varying order of thinness , J. Dynam. Differential Equations, 10 (1998), 73–108 MR1607535 10.1023/A:1022640429041
\ref \no \dfa HaRau1..J. Hale and G. Raugel, Reaction-diffusion equations on thin domains , J. Math. Pures Appl. (9), 71 (1992), 33–95 \ref \no \dfa HaRau2.. ––––, A damped hyperbolic equation on thin domains , Trans. Amer. Math. Soc., 329 (1992), 185–219 \ref \no \dfa HaRau3..––––, A reaction-diffusion equation on a thin $L$-shaped domain , Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283–327 \ref \no \dfa HaRau1..J. Hale and G. Raugel, Reaction-diffusion equations on thin domains , J. Math. Pures Appl. (9), 71 (1992), 33–95 \ref \no \dfa HaRau2.. ––––, A damped hyperbolic equation on thin domains , Trans. Amer. Math. Soc., 329 (1992), 185–219 \ref \no \dfa HaRau3..––––, A reaction-diffusion equation on a thin $L$-shaped domain , Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283–327
\ref\no \dfa He.. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture notes in mathematics, 840 , Springer–Verlag, New York (1981) MR610244\ref\no \dfa He.. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture notes in mathematics, 840 , Springer–Verlag, New York (1981) MR610244
\ref \no \dfa priry..M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations , Studia Math., 151 (2002), 109-140 MR1917228 10.4064/sm151-2-2\ref \no \dfa priry..M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations , Studia Math., 151 (2002), 109-140 MR1917228 10.4064/sm151-2-2
\ref \no \dfa pr..M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 173 (2001), 271–320 \ref \no \dfa pr2.. ––––, Inertial manifolds on squeezed domains , J. Dynam. Differential Equations, to appear \ref \no \dfa pr3.. ––––, Some recent results on thin domain problems, Topol. Methods Nonlinear Anal., 14(1999), 239–255 MR1834117 10.1006/jdeq.2000.3917\ref \no \dfa pr..M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 173 (2001), 271–320 \ref \no \dfa pr2.. ––––, Inertial manifolds on squeezed domains , J. Dynam. Differential Equations, to appear \ref \no \dfa pr3.. ––––, Some recent results on thin domain problems, Topol. Methods Nonlinear Anal., 14(1999), 239–255 MR1834117 10.1006/jdeq.2000.3917
\ref\no \dfa Rau1.. G. Raugel, Dynamics of Partial Differential Equations on Thin Domains (R. Johnson, ed.), Dynamical systems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13–22, 1994. Lect. Notes in Math. Vol. 1609, Springer–Verlag, Berlin (1995), 208–315 MR1374110 10.1007/BFb0095241\ref\no \dfa Rau1.. G. Raugel, Dynamics of Partial Differential Equations on Thin Domains (R. Johnson, ed.), Dynamical systems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13–22, 1994. Lect. Notes in Math. Vol. 1609, Springer–Verlag, Berlin (1995), 208–315 MR1374110 10.1007/BFb0095241
\ref\no \dfa rav.. P. A. Raviart and J. M. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson, Paris(1983) MR773854\ref\no \dfa rav.. P. A. Raviart and J. M. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Masson, Paris(1983) MR773854
\ref\no \dfa wein.. Hans F. Weinberger, Variational Methods for Eigenvalue Approximation, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia(1974) MR400004\ref\no \dfa wein.. Hans F. Weinberger, Variational Methods for Eigenvalue Approximation, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia(1974) MR400004
