Local behaviour of solutions to doubly nonlinear parabolic equations (Q877166)

Nonnegative weak solutions to the doubly nonlinear parabolic equation \[ \text{div} (| Du| ^{p-2}Du)=\partial_t (u^{p-1}), \] \(p\in (1, \infty),\) are considered. The main purpose of the article is to give a relatively simple and transparent proof for Harnack's inequality using the approach of Moser. To show that the proof is based on a general principle the Lebesgue measure is replaced with an arbitrary doubling Borel measure which supports a Poincaré inequality. The arguments can be applied to more general equations with replacement of \(p\)-Laplacian by divergent operator \(\operatorname{div} A(x,t,u,Du)\) with standard structure conditions.