Gradual loss of positivity and hidden invariant cones in a scalar heat equation

Abstract

Invariance properties of a scalar, linear heat equation with nonlocal boundary conditions are discussed as a function of a real parameter appearing in the boundary conditions of the problem. The equation is a model for a thermostat with sensor and controller positioned at opposite ends of an interval, whence the non-locality. It is shown that the analytic semigroup associated with the evolution problem is positive if and only if the parameter is in $(-\infty,0]\,$. For the corresponding elliptic problem three maximum principles are proved which hold for different parameter ranges.