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Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. (English) Zbl 0735.35082
The authors construct unique global continuous viscosity solutions of the initial value problem in \(\mathbb{R}^ n\) for a class of degenerate parabolic equations that they call geometric. A typical example is \[ u_ t- |\nabla u|\text{div}(\nabla u/|\nabla u|)-\nu|\nabla u|=0. (*) \] For a solution \(u\), let \(\Gamma(t)=\{x\in\mathbb{R}^ n: u(x,t)=\gamma\}\) and \(D(t)=\{x\in\mathbb{R}^ n: u(x,t)>\gamma\}\). Then the family \((\Gamma(t),D(t))\), \(t\geq0\), is uniquely determined by \((\Gamma(0),D(0))\) and is independent of \(u\) and \(\gamma\). Moreover \((\Gamma(t),D(t))\) becomes empty in a finite time provided \(\nu\leq0\). This extends a result of G. Huisken [J. Differ. Geom. 20, 237-266 (1984; Zbl 0556.53001)]. The main tool is a comparison principle for viscosity solutions, but proofs are not given.
Reviewer: A.Pryde (Clayton)

MSC:
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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