Global existence and blow-up of solutions to porous medium equation for Baouendi-Grushin operator

TL;DR

This paper investigates conditions under which solutions to a nonlinear porous medium equation involving the Baouendi-Grushin operator either exist globally or blow up, using concavity and Poincaré inequalities.

Contribution

It establishes new results on global existence and blow-up for solutions to the porous medium equation with Baouendi-Grushin operator, extending previous analytical techniques.

Findings

Identifies conditions for global existence of solutions.

Determines criteria for finite-time blow-up.

Utilizes concavity and Poincaré inequalities in the analysis.

Abstract

In this note, we show a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation related to Baouendi-Grushin operator. Our approach is based on the concavity argument and the Poincar\'e inequality for Baouendi-Grushin vector fields from [35], inspired by the recent works [28] and [29].