Global existence and finite time blowup for a mixed pseudo-parabolic $p$-Laplacian type equation

TL;DR

This paper investigates the conditions under which solutions to a mixed pseudo-parabolic p-Laplacian equation exist globally or blow up in finite time, providing explicit criteria and analyzing different energy regimes.

Contribution

It introduces a family of potential wells, derives explicit potential depth, and establishes existence, uniqueness, decay, and blowup results across subcritical, critical, and supercritical initial energies.

Findings

Global solutions exist with decay estimates for subcritical energy.

Solutions blow up in finite time for certain initial energies.

Extended results to critical and supercritical energy cases.

Abstract

This paper concerns the initial-boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then prove the existence, uniqueness and decay estimate of global solutions and the blowup phenomena of solutions with subcritical initial energy. Next, we extend parallelly these results to the critical initial energy. Lastly, the existence, uniqueness and asymptotic behavior of global solutions with supercritical initial energy are proved by further analyzing the properties of $\omega$-limits of solutions.