Shape-preserving signal forms in heat conduction

Let us consider the heat conduction problem described by a parabolic equation. We study under which conditions is the time-dependence on the boundary preserved inside the solid. The question is how information entering on the boundary penetrates the solid. E.g. consider a heat conducting solid subject to sinusoidally varying boundary condition. After decay of the transients, the temperature at any inner point varies in time sinusoidally with the same circular frequency, with space dependent amplitude and phase delay. So, sinusoidal signals inserted on the boundary are preserved. Information is also preserved in case of linear signals. Farkas and Mudri H. Farkas, I. Mudri, Shape-preserving time-dependences in heat conduction, Acta Phys. Hung. 55 (1984) 267–273] have formulated this phenomenon, defined the notion of the boundary following solution and the shape-preserving signal forms, determined necessary and heuristic sufficient conditions for the shape-preserving signal forms.

Their work is extended by rigorous proofs of some sufficient conditions in this paper, and the minimum of the phase delay, expected to be attained on the boundary for physical reasons, is examined.