Floquet Bundles for Scalar Parabolic Equations

摘要

For linear scalar parabolic equations such as Ut = Uxx + a(t,x)Ux + b(t,x)U on afinite interval 0 <= x <= pi, with various boundary conditions, we obtain canonical Floquet solutions Un(t,X). These solutions are characterized by the property that Z(Un(t, x)) = n for all t in R, where z(.) denotes the zero crossing (lap) number of Matano. The coefficients a(t,x) and b(t,x) are not assumed to be periodic in t, but if they are, the solutions Un(t,X) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N >= 1 an exponential dichotomy between the bundles span.Un(.,.)N sub = 1 and span.Un(.,.)infinity sub n = N + 1.