A note on comparison principle for \(p\)-Laplacian evolution type equation. (English) Zbl 1470.35085

Guidolin, P. L.; Schütz, L.; Ziebell, J. S.

J. Elliptic Parabol. Equ. 7, No. 1, 65-73 (2021).

Summary: In this paper we provide a comparison principle for the weak solutions \(u(\cdot ,t), v(\cdot ,t)\) of two similar evolution \(p\)-Laplacian equations, both with source terms in a divergent and non-divergent form. Once we treat with signal solutions defined in all space \(\mathbb{R}^n\), for all \(t\) in a maximal existence interval \([0,T_{\ast})\), the arguments presented here differ from the ones used to prove the comparison principle in bounded domains. We suppose \(p \geq n, p>2\) and also consider some additional natural assumptions. The initial conditions \(u(\cdot ,0)\) and \(v(\cdot ,0)\) are supposed to belong to the space \(L^1(\mathbb{R}^n) \cap L^{\infty }(\mathbb{R}^n)\). An useful proposition to prove the comparison principle will be presented and the contraction of the \(L^1\) norm of \(u-v\) for a particular case will be shown.

MSC:

35B51	Comparison principles in context of PDEs
35K15	Initial value problems for second-order parabolic equations
35K59	Quasilinear parabolic equations
35K92	Quasilinear parabolic equations with \(p\)-Laplacian

Keywords:

source terms in a divergent and non-divergent form

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