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Gradual loss of positivity and hidden invariant cones in a scalar heat equation. (English) Zbl 0983.35013
Authors’ 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 nonlocality. 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.

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 47D06 One-parameter semigroups and linear evolution equations 35K20 Initial-boundary value problems for second-order parabolic equations 35B50 Maximum principles in context of PDEs