Tohoku Mathematical Journal
2018
September
SECOND SERIES VOL. 70, NO. 3

Tohoku Math. J.
70 (2018), 425-445

Title THE EQUIVALENCE OF WEAK AND VERY WEAK SUPERSOLUTIONS TO THE POROUS MEDIUM EQUATION

Author Pekka Lehtelä and Teemu Lukkari

(Received December 18, 2015, revised March 14, 2016)
Abstract. We prove that various notions of supersolutions to the porous medium equation are equivalent under suitable conditions. More spesifically, we consider weak supersolutions, very weak supersolutions, and $m$-superporous functions defined via a comparison principle. The proofs are based on comparison principles and a Schwarz type alternating method, which are also interesting in their own right. Along the way, we show that Perron solutions with merely continuous boundary values are continuous up to the parabolic boundary of a sufficiently smooth space-time cylinder.

Mathematics Subject Classification. Primary 35K65, Secondary 35K20, 35D30, 35D99, 31C45.
Key words and phrases. Porous medium equation, weak solutions, very weak solutions, supersolutions, comparison principle, boundary value problems.

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