  We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t
u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the
study of extremals of Poincar\'e inequalities in Sobolev spaces. We prove
spatial Lipschitz continuity and H\"older continuity in time of order $(p-1)/p$
for viscosity solutions. As an application of our estimates, we obtain
pointwise control of the large time behavior of solutions.
