Phragmèn-lindelÖf type results for periodic solutions of parabolic equations on a half-line

Abstract

Let u be a time periodic solution of a parabolic equation with time periodic coefficients on the half-line x > 0. We prove that if ⨿u(x,·)⨿ ⩽ exp(γx) for a certain nerm, where γ > γo, then u ( x , t ) is bounded. The constant γo makest he following sense. Let the equation be written as ane volution equation in the space of periodic 2-vector function s( u,ux). Then the evolution operator has discrete spectrum, and γ0 is the minimal positive real part of all the eigenvalues