Parabolic problems in highly oscillating thin domains

Abstract

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 region of R 2 that degenerates into a line segment when a positive parameter 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 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. 2010 Mathematics Subject Classification. 35R15, 35B27, 35B40, 35B41, 35B25, 74Q10. Key words and phrases. Partial differential equations on infinite-dimensional spaces, asymptotic behavior of solutions, attractors, singular perturbations, thin domains, oscillatory behavior, lower semicontinuity, homogenization. † Partially supported by CNPq 302847/2011-1, CAPES/DGU 267/2008 and FAPESP 2008/53094-4, Brazil.

Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary

“…As we will see and as it is already well known, see for instance [4,5,8,35], the analysis of the convergence of (2) will basically dictate the behavior of the dynamics of (3) in terms of the behavior of the global dynamics (continuity of solutions, continuity of equilibria, upper and lower semicontinuity of attractors, etc). As a matter of fact, if the solutions of (2) approach in certain sense the solutions of the limiting problem −Lw + w = f in ω, where L will be an elliptic operator then, the solutions of (3) will converge in certain sense to the solutions of the equation w t − Lw + w = f (w) in ω.…”

Numerical simulation of high sensitivity of solutions to the nonlinear singularly perturbed dynamical system on the initial conditions and parameter

“…The fact that the oscillations occur in diverse phenomena of mechanics, chemistry, biology, medicine and even economics and other areas, makes their control to be an interesting research question. Recently, non classical relaxation oscillations have been studied in the Olsen model for the peroxidase-oxidase reaction [11] and asymptotic behavior of the solutions of a semilinear parabolic problem with homogeneous Neumann boundary condition in a thin domain exhibiting highly oscillatory behavior has been considered in [14]. Slow-fast dynamics of tri-neuron Hopfield neural network with two time scales with relaxation oscillations was investigated in [20], proving that the Hopfield neural network has the potential to reproduce the complex dynamics of real neurons.…”

Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

“…In [18] the authors consider nonlinear monotone problems in a multidomain with a highly oscillating boundary. In [19,20,21,22,23,24,25,26] and references therein, we have recently studied many classes of oscillating thin regions for elliptic and parabolic equations with Neumann boundary conditions, discussing limit problems and convergence properties. For nonlocal equations in thin structures we also mention [27,28,29,30].…”