Tohoku Mathematical Journal

The equivalence of weak and very weak supersolutions to the porous medium equation

Pekka Lehtelä and Teemu Lukkari

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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.

Article information

Source
Tohoku Math. J. (2), Volume 70, Number 3 (2018), 425-445.

Dates
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1537495355

Digital Object Identifier
doi:10.2748/tmj/1537495355

Mathematical Reviews number (MathSciNet)
MR3856775

Zentralblatt MATH identifier
06996536

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 35D30: Weak solutions 35D99: None of the above, but in this section 31C45: Other generalizations (nonlinear potential theory, etc.)

Keywords
porous medium equation weak solutions very weak solutions supersolutions comparison principle boundary value problems

Citation

Lehtelä, Pekka; Lukkari, Teemu. The equivalence of weak and very weak supersolutions to the porous medium equation. Tohoku Math. J. (2) 70 (2018), no. 3, 425--445. doi:10.2748/tmj/1537495355. https://projecteuclid.org/euclid.tmj/1537495355


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