an:03875840
Zbl 0549.35061
DiBenedetto, Emmanuele; Friedman, Avner
H??lder estimates for non-linear degenerate parabolic systems
EN
J. Reine Angew. Math. 357, 1-22 (1985).
00149634
1985
j
35K55 35K65 35B45
weak solution; nonlinear degenerate parabolic system; H??lder continuous; scaling; space-time cylinders
It is shown that for any weak solution of the nonlinear degenerate parabolic system
\[
\partial u^ i/\partial t-div(|\nabla u|^{p-2}\nabla u^ i)=F_ i(x,t,\nabla u)\quad (i=1,...,m)
\]
in \(\Omega\times (0,T)\) when \(u=(u^ 1,...,u^ m)\), \(\Omega\subset {\mathbb{R}}^ n\), \(p>\max\{1,2N/(N+2)\},\) the spatial gradient \(\nabla u\) is H??lder continuous provided \(| F_ i(x,t,\nabla u)|\leq C|\nabla u|^{p-1}+f_ i(x,t),\quad f_ i\in L^ q,\quad q>pN/(p-1).\) The proof is based on the method of a previous paper by the authors in J. Reine Angew. Math. 349, 83-128 (1984; Zbl 0527.35038), but employs a new scaling (for the space-time cylinders) which reflects the degeneracy in the system. Using this type of scaling, it is also proved that the solution of the degenerate porous medium equation for anisotropic material is H??lder continuous.
Zbl 0527.35038