Local and global boundedness for some nonlinear parabolic equations exhibiting a time singularity

Abstract

Results on the local and global boundedness of nonnegative weak subsolutions of the doubly nonlinear parabolic equation $$ (u^{q})_t-\text{div}\,{(|\nabla u|^{p-2}\nabla u)}=0, $$ are obtained for $p > 1$ and $0 < q < 1$, that is, for equations presenting a singularity in the time derivative part (as well as a singularity, $1 < p < 2$, or degeneracy, $p > 2$, in the principal part of the operator). We work in measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality.