Floquet theory for parabolic differential equations
The Floquet theorem is extended to the scalar parabolic equation \[ u_ t = u_{xx} + a(t,x) u_ x + b(t,x)u, \quad 0<x< \pi \] with Dirichlet boundary conditions \(u(t,0) = (t, \pi) = 0\) on the interval \((0,\pi)\). The coefficients \(a(t,x)\), \(b(t,x)\) are assumed to be \(T\)-periodic in \(t\) for all \(x\), and sufficiently smooth. Considering this problem as a system in the Hilbert space \(H=L^ 2(0,\pi)\) the existence of a time- dependent linear isomorphism \(v=\Phi(t)u\) is proved which is \(T\)-periodic as well as continuous in \(t\) in the operator norm and which transforms this problem to the autonomous system \[ v_ t = v_{xx} + q(x)v, \quad v(t,0) = v(t,\pi) = 0 \] for some function \(q(x)\).
scientific article; zbMATH DE number 589984