, Volume 145, Issue 1, pp 385-405

On 385-01385-01385-01regularity of the gradient of solutions of degenerate parabolic systems

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Summary

We consider weak solutions u∈Lp([0,T], W p 1 (Ω))∩([0,T],L2(Ω))of the degenerate parabolic (model) -system $$\frac{{\partial u^i }}{{\partial t}} - div (|\nabla u|^{p - 2} \nabla u^i ) = 0 on \Omega \subset R^N , 1 \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} i \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} m with p > 2.$$ By local techniques it is proved, using sequences of time-space cylinders, which are adjusted to the alternative whether one is at a point of degeneracy or not, that the spatial gradient of u is α- Höldercontinuous on compact subsets of Ω× [0,T] with some α which depends only on N and p.