Homogenization limit of a parabolic equation with nonlinear boundary conditions

Consider the parabolic equation u(t) = a(u(x))u(xx) + f (u(x)), -1 < x < 1, t > 0, (E) with nonlinear boundary conditions: u(x)(-1, t) = g(u(-1, t)/epsilon), u(x)(1, t) = -g(u(1, t)/epsilon), (NBC) where epsilon > 0 is a parameter, g is a function which takes values near its supremum "frequently". Each almost periodic function is a special example of g. We consider a time-global solution u(epsilon) of (E)-(NBC) and show that its homogenization limit as epsilon -> 0 is the solution eta of (E) with linear boundary conditions: eta(x)(-1, t) = sup g, eta(x)(1, t) = -sup g, (LBC) provided eta moves upward monotonically. When g is almost periodic, Lou (preprint) [21] obtained the (unique) almost periodic traveling wave U(epsilon) of (E)-(NBC). This paper proves that the homogenization limit of U(epsilon) is a classical traveling wave of (E)-(LBC). (C) 2011 Elsevier Inc. All rights reserved.