Electron. J. Differential Equations, Vol. 2022 (2022), No. 08, pp. 1-17.

Blow-up for parabolic equations in nonlinear divergence form with time-dependent coefficients

Xuhui Shen, Juntang Ding

Abstract:
In this article, we study the blow-up of solutions to the nonlinear parabolic equation in divergence form,

where $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ $(n\geq2)$ with smooth boundary $\partial\Omega$ . By constructing suitable auxiliary functions and using a differential inequality technique, when $\Omega\subset\mathbb{R}^n$ $(n\geq2)$, we establish conditions for the solution blow up at a finite time, and conditions for the solution to exist for all time. Also, we find an upper bound for the blow-up time. In addition, when $\Omega\subset \mathbb{R}^n$ with $(n\geq3)$, we use a Sobolev inequality to obtain a lower bound for the blow-up time.

Submitted January 27, 2021. Published January 25, 2022.
Math Subject Classifications: 35K55, 35B44.
Key Words: Nonlinear parabolic equation; blow-up; upper bound; lower bound.
DOI: https://doi.org/10.58997/ejde.2022.08

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Xuhui Shen
School of Mathematical Sciences
Shanxi University
Taiyuan 030006, China
email: xhuishen@163.com
Juntang Ding
School of Mathematical Sciences
Shanxi University
Taiyuan 030006, China
email: djuntang@sxu.edu.cn

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