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On breakdown of solutions of a constrained gradient system of total variation. (English) Zbl 1064.35084

Summary: A gradient system of total variation of the type \[ u_t= \text{div}\left (\frac{\nabla u}{|\nabla u|}\right)+|\nabla u|u, \] is considered for a mapping from the unit disk to the unit sphere in \(\mathbb{R}^3\). For a class of initial data it is shown that a solution of its Dirichlet problem loses its smoothness in finite time.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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