# Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

## TL;DR

This paper investigates the initial boundary problem for a quasilinear parabolic equation, classifying blowup and extinction phenomena based on reaction exponents using inequalities, energy methods, and comparison principles.

## Contribution

It provides a complete classification of blowup and extinction phenomena for the equation across different reaction exponent ranges.

## Key findings

- Complete classification of blowup phenomena.
- Conditions for extinction of solutions.
- Analysis based on reaction exponents.

## Abstract

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{\beta-1}u+\alpha|u|^{q-2}u, \] where $p>1,\beta>0$, $q\geq1$ and $\alpha>0$. By using Gagliardo-Nirenberg type inequality, energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.