This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is
not
asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period
T
of the boundary data with the square of the length of the interval over
π
.
The Lp−Lp′ estimate for the Schrödinger equation on the half-line
