# A Parabolic Equation with a Mean-Curvature Type Operator

Chapter

## Abstract

The classical mean-curvature operator arises in geometry, and is given by. Mathematically this operator is of particular interest because of its degeneracy for large gradients. The same type of operators arises also in thermodynamical context in the theory of convex free energy functionals, which are asymptotically linear as the gradient tends to infinity [12]; in particular this leads to the parabolic equation *u* _{ t } = *Lu* [6]. Observe that here *u* is supposed to be a *function*, i.e. *u* is single-valued, in contrast to the geometric origin of the operator *L*, where one considers *surfaces* which may correspond very well to multi-valued functions *u*.

## Keywords

Shock Wave Parabolic Equation Entropy Solution Entropy Condition Ground State Solution
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