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
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.
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Funding was provided by Fundação para a Ciância e a Tecnologia.
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Research supported by the project PTDC/MAT/CAL/0749/2012.
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Henriques, E., Laleoglu, R. Boundedness for Some Doubly Nonlinear Parabolic Equations in Measure Spaces. J Dyn Diff Equat 30, 1029–1051 (2018). https://doi.org/10.1007/s10884-017-9585-3
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DOI: https://doi.org/10.1007/s10884-017-9585-3