THE LEFSCHETZ DIRECT STABILITY CRITERION FOR AN IMPLICIT EVOLUTION PROBLEM, WITH A DYNAMIC BOUNDARY CONDITION

We consider a nonlinear dynamic problem comprising a system of nonlinear parabolic equations. The second of these equations is a nonlinear dynamic boundary condition to the problem. The derivation of the system is through the heat energy conservation laws [4]. Problems of this type occur in heat energy absorption and release through the surfaces of solids. The second equation to the system describes surface radiation itself. We rewrite the system as an implicit evolution equation, thus exposing trace-like canonical operators. These operators have been studied and characterized in [3] and [4]. Subsequent to that, we study the stability of the null solution to the implicit evolution problem using the modified Lefschetz [6] system for the direct stability criterion. We show that, even though the modified Lefschetz system leads to a new Lyapunov function for the problem, the Lefschetz direct stability criterion itself is invariant. We test the modified Lefschetz system on the cooling problem in [3], by constructing the corresponding Lyapunov function and confirming its known properties. Symbols used: 1. alpha, beta, alpha(0) and beta(0) are positive real constants; 2. Omega is an open bounded domain in R-3, where partial derivative Omega not subset of Omega; 3. Delta := del . del; 4. Delta(s) = del(s) . del(s); the Beltrammi-Laplace operator; with del(s) := partial derivative/partial derivative s(1)tau(1) + partial derivative/partial derivative s(2)tau(2) for an arbitrary point (tau(1), tau(2)) on partial derivative Omega.