Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator

TL;DR

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

Contribution

It introduces a new analysis framework for the pseudo-parabolic equation with the Baouendi-Grushin operator, combining concavity arguments and Poincaré inequalities.

Findings

Established criteria for global existence of solutions.

Identified conditions leading to blow-up of solutions.

Extended previous methods to a new class of operators.

Abstract

In this note, we study a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear pseudo-parabolic equation for the Baouendi-Grushin operator. The approach is based on the concavity argument and the Poincar\'e inequality related to the Baouendi-Grushin operator from [24], inspired by the recent work [21].