Remarks on a paper by W. A. Day on a maximum principle under nonlocal boundary conditions

In a recent paper W. A. Day proved a decay property for solutions to a linear parabolic equation with nonlocal boundary conditions. Such boundary conditions arise in thermoelasticity. We extend his result (a) from one to arbitrary space dimensions, (b) from linear to nonlinear parabolic equations, and (c) from differential equations to differential inequalities. Our tool is the classical maximum principle.