Boundedness for some doubly nonlinear parabolic equations in measure spaces (Q1994001)

The authors derive interesting local and global sup bounds of the nonnegative weak subsolutions of \[ (u^q)_t-\nabla\cdot(|\nabla u|^{p-2}\nabla u)=0, \] in \(U_\tau=U\times(\tau_1,\tau_2],\quad p>1,\quad q>1\) and of its associated Dirichlet problem. All these results are obtained in measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality. For particular ranges of the exponents \(p\) and \(q\), the authors show that any locally nonnegative weak subsolution, taken in \(Q\subset\bar Q\subset U\tau\), is controlled from above by the \(L^\alpha(\bar Q)\)-norm, for \(\alpha=\max\{p,q+1\}\). As for the global setting, under suitable assumptions on the boundary datum \(g\) and on the initial datum, the authors obtain sup bounds for \(u\), in \(U\times\{t\}\), which depend on the sup of \(g\) and on the \(L^{q+1}(U\times(\tau_1,\tau_1+t])\)-norm of \((u-\sup g)_+\), for all \(t\in (0,\tau_2-\tau_1]\). On the critical ranges of \(p\) and \(q\), as expected, a priori local and global \(L^\infty\) estimates require extra qualitative information on \(u\).