## Tohoku Mathematical Journal

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

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

Pekka Lehtelä and Teemu Lukkari

#### 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