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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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##### References:
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