A vector valued function $u=u(x,t)$, solution of a quasilinear parabolic system cannot be too close to a straight line without being regular.
@article{118862, author = {Eugen Viszus}, title = {A note on regular points for solutions of parabolic systems}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {37}, year = {1996}, pages = {557-563}, zbl = {0881.35055}, mrnumber = {1426920}, language = {en}, url = {http://dml.mathdoc.fr/item/118862} }
Viszus, Eugen. A note on regular points for solutions of parabolic systems. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 557-563. http://gdmltest.u-ga.fr/item/118862/
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