Giga, Yoshikazu; Kuroda, Hirotoshi On breakdown of solutions of a constrained gradient system of total variation. (English) Zbl 1064.35084 Bol. Soc. Parana. Mat. (3) 22, No. 1, 9-20 (2004). 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. Cited in 6 Documents MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs Keywords:Dirichlet problem PDFBibTeX XMLCite \textit{Y. Giga} and \textit{H. Kuroda}, Bol. Soc. Parana. Mat. (3) 22, No. 1, 9--20 (2004; Zbl 1064.35084)