Opuscula Math. 38, no. 1 (2018), 117-131
https://doi.org/10.7494/OpMath.2018.38.1.117

Opuscula Mathematica

Study of ODE limit problems for reaction-diffusion equations

Jacson Simsen
Mariza Stefanello Simsen
Aleksandra Zimmermann

Abstract. In this work we study ODE limit problems for reaction-diffusion equations for large diffusion and we study the sensitivity of nonlinear ODEs with respect to initial conditions and exponent parameters. Moreover, we prove continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction-diffusion equations with spatially variable exponents when the exponents go to 2 in $$L^{\infty}(\Omega)$$ and the diffusion coefficients go to infinity.

Keywords: ODE limit problems, shadow systems, reaction-diffusion equations, parabolic problems, variable exponents, attractors, upper semicontinuity.

Mathematics Subject Classification: 35B40, 35B41, 35K57, 35K59.

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• Jacson Simsen
• Universidade Federal de Itajubá, Instituto de Matemática e Computação, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajubá - MG - Brazil
• Universität of Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany
• Mariza Stefanello Simsen
• Universidade Federal de Itajubá, Instituto de Matemática e Computação, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajubá - MG - Brazil
• Universität of Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany
• Aleksandra Zimmermann
• Universität of Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany
• Communicated by Bruno Welfert.
• Revised: 2017-07-10.
• Accepted: 2017-08-01.
• Published online: 2017-11-13.