Full regularity for a class of degenerated parabolic systems in two spatial variables

The authors consider quasilinear parabolic systems in two space dimensions. The function a has p-growth behaviour, 1< p < ∞, and the ellipticity “constant” behaves like (1+|∇u|) ^{p − 2}. The author prove full regularity of the weak solution on interior subdomains, but globally in time. The key idea in the proof is a technique to obtain boundedness of the gradient based on logarithmic estimates. Received: 8 July 1998 / Revised version: 19 February 1999