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Global behaviour of solutions to a parabolic mean curvature equation. (English) Zbl 0844.35050
The authors investigate the global behaviour of solutions of $u_t- \text{div}\Biggl({Du\over \sqrt{1+|Du|^2}}\Biggr)= 1\quad\text{in}\quad \Omega\times (0, \infty)$ in the case that there is no stationary solution. In this case solutions may blow up as $$t\to \infty$$ and may detatch from their boundary values. The authors use the theory of viscosity solutions to investigate these phenomena in detail in the special cases that $$\Omega$$ is either a ball in $$\mathbb{R}^n$$ or a square in $$\mathbb{R}^2$$. They verify certain conjectures of P. Marcellini and K. Miller [J. Differ. Equ. 51, 326-358 (1984; Zbl 0545.35044)] concerning the relation between solutions of the above problem and a certain isoperimetric problem.
Reviewer: J.Urbas (Bonn)

MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000) 49Q10 Optimization of shapes other than minimal surfaces 35R35 Free boundary problems for PDEs