Homogenization limit of a parabolic equation with nonlinear boundary conditions☆

Publication year: 2011
Source: Journal of Differential Equations, In Press, Corrected Proof, Available online 14 June 2011

Bendong, Lou

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