Semilinear parabolic problems in thin domains with a highly oscillatory boundary

• Published in: Nonlinear analysis Vol. 74; no. 15; pp. 5111 - 5132
• Main Authors: Arrieta, José M., Carvalho, Alexandre N., Pereira, Marcone C., Silva, Ricardo P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.10.2011; Elsevier
• Subjects: Boundaries; Convergence; Dissipative parabolic equations; Dynamical systems; Exact sciences and technology; Global attractors; Homogenization; Homogenizing; Lower semicontinuity; Mathematical analysis; Mathematics; Nonlinear dynamics; Nonlinearity; Partial differential equations; Sciences and techniques of general use; Segments; Thin domains; Upper semicontinuity; Homogenization; Upper semicontinuity; Lower semicontinuity; Thin domains; Global attractors; Dissipative parabolic equations; Parabolic equation; Nonlinear analysis; Mathematical model; Partial differential equation; Global attractor
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.05.006

In this paper, we study the behavior of the solutions of nonlinear parabolic problems posed in a domain that degenerates into a line segment (thin domain) which has an oscillating boundary. We combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problems and analyze the convergence properties of the solutions and attractors of the evolutionary equations.