The time derivative in a singular parabolic equation. (English) Zbl 1424.35235

Lindqvist, Peter

Differ. Integral Equ. 30, No. 9-10, 795-808 (2017).

In this paper, the author considers the \(p\)-Laplacean evolution equation \[u_t=\Delta_pu\] with \(1<p<2\) and he is able to prove that the solutions have a first order time derivative \(u_t\) in Sobolev’s sense. This is very interesting because, by using the celebrated methods by DeGiorgi, Nash, and Moser, it is possible to prove the regularity of the space derivatives but it is impossible to treat directly the time derivative which is regarded as merely a distribution.

Reviewer: Vincenzo Vespri (Firenze)

MSC:

35K67	Singular parabolic equations
35K92	Quasilinear parabolic equations with \(p\)-Laplacian
35B45	A priori estimates in context of PDEs

Keywords:

\(p\)-Laplacean evolution operator; singular case; regularity of the time derivative