Extensions of a property of the heat equation to linear thermoelasticity and other theories

is a decreasing function of t for 0 < t < T. It should be noted that (P) holds whatever initial values the temperature may take on f = 0. The adjective 'decreasing' is to be understood in the wide sense as meaning 'monotone nonincreasing'. The condition 9(x, t) e C2(R) can easily be replaced by one that is weaker, but to insist upon the weakest smoothness hypotheses for our theorems would overburden them and obscure their main point and instead we shall habitually assume more then is really necessary. Property (P) was formulated and proved first by Polya and Szego [1]. It is in fact a straightforward deduction from the maximum principle but a different method of proof, based upon convexity arguments, was discovered by Bellman [2] and it turns out that it is Bellman's method which is the more suited to proving the extensions we have in mind. The heat equation describes the conduction of heat with considerable success. From the point of view of continuum mechanics, though, it rests upon highly restrictive assumptions, and it is interesting to ask if (P), or some suitably modified form of (P), continues to hold in other theories which reflect more nearly the behavior of real bodies. The object of this paper is to show that modified forms of (P) remain true within both the quasi-static and the dynamic theories of coupled linear thermoelasticity—these theories remove the rigidity requirement—and also within certain theories which replace the parabolic heat equation by a linear hyperbolic equation and thereby ensure that temperature disturbances propagate at finite speed.