Floquet bundles for scalar parabolic equations

Abstract

For linear scalar parabolic equations such as $$u_t = u_{xx} + a(t,x)u_x + b(t,x)u$$ on a finite interval 0≦x≦π, with various boundary conditions, we obtain canonical Floquet solutions u n (t, x). These solutions are characterized by the property that z(u n (t, x))=n for all teℝ, 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 u n (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 {u n (·,·)} =1/ and $$\overline {span} \{ u_n ( \cdot , \cdot )\} _{n = N + 1}^\infty $$ .