Homogenization limit of a parabolic equation with nonlinear boundary conditions

Consider the parabolic equationequation(E)ut=a(ux)uxx+f(ux),−10, with nonlinear boundary conditions:ux(−1,t)=g(u(−1,t)/ε),ux(−1,t)=g(u(−1,t)/ε),equation(NBC)ux(1,t)=−g(u(1,t)/ε),ux(1,t)=−g(u(1,t)/ε), where ε>0ε>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 uεuε of (E)–(NBC) and show that its homogenization limit as ε→0ε→0 is the solution η of (E) with linear boundary conditions:equation(LBC)ηx(−1,t)=supg,ηx(1,t)=−supg, provided η moves upward monotonically. When g is almost periodic, Lou (preprint) [21] obtained the (unique) almost periodic traveling wave UεUε of (E)–(NBC). This paper proves that the homogenization limit of UεUε is a classical traveling wave of (E)–(LBC).