Differential and Integral Equations

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

Patrick Guidotti and Sandro Merino

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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.

Article information

Differential Integral Equations, Volume 13, Number 10-12 (2000), 1551-1568.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35B10: Periodic solutions 35B32: Bifurcation [See also 37Gxx, 37K50] 35B50: Maximum principles 35J25: Boundary value problems for second-order elliptic equations 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Guidotti, Patrick; Merino, Sandro. Gradual loss of positivity and hidden invariant cones in a scalar heat equation. Differential Integral Equations 13 (2000), no. 10-12, 1551--1568. https://projecteuclid.org/euclid.die/1356061139

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