2013/12/04 by Marcone C. Pereira, Pereira, Marcone C.
Computer Science · Engineering · Mathematics · #35B25 #35B27 #35B40 #35B41 #35R15 #74Q10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #msc:35B25 #msc:35B27 #msc:35B40 #msc:35B41 #msc:35R15 #msc:74Q10
paper · pdf · doi:10.48550/arxiv.1312.1131
41 pages, 2 figures
arxiv created 2013/12/04 · openalex publication_date 2013/12/04 · arxiv updated 2013/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a bounded region of the plane that degenerates into a line segment when a positive parameter ε goes to zero (a thin domain). Here we also allow that its boundary presents highly oscillatory behavior with different orders and variable profile. We take thin domains possessing the same order ε to the thickness and amplitude of the oscillations but assuming different order to the period of oscillations on the top and the bottom of the boundary. We combine methods from linear homogenization theory and the theory on nonlinear dynamics of dissipative systems to obtain the limit problem establishing convergence properties for the solutions. At the end we show the upper semicontinuity of the attractors and stationary states.