Boundedness for Some Doubly Nonlinear Parabolic Equations in Measure Spaces

Abstract

In the context of measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality, we derive local and global sup bounds of the nonnegative weak subsolutions of

$$\begin{aligned} (u^{q})_t-\nabla \cdot {(|\nabla u|^{p-2}\nabla u)}=0, \quad \mathrm {in} \ U_\tau = U \times (\tau _1, \tau _2] , \quad p>1,\quad q>1 \end{aligned}$$

and of its associated Dirichlet problem, respectively. For particular ranges of the exponents p and q, we 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, we obtain sup bounds for u, in $U \times \{ t\}$, which depend on the $\sup g$ 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, a priori local and global $L^\infty$ estimates require extra qualitative information on u.