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March/April 2019 Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption
Yingying Liu, Zhengce Zhang, Liping Zhu
Adv. Differential Equations 24(3/4): 229-256 (March/April 2019).

Abstract

This paper deals with a quasilinear parabolic equation with nonlinear gradient absorption \begin{equation*} u_t-\Delta_{p}u=u^q-\mu u^{r}|\nabla u|^\delta, \ x\in\Omega, t>0. \end{equation*} Here, $\Delta_{p} u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplace operator, $\Omega \subset \mathbb{R}^{N}$ $(N \geq 1)$ is a bounded smooth domain. By a regularization approach, we first establish the local-in-time existence of its weak solutions. Then we prove the global existence by constructing a family of bounded super-solutions which technically depend on the inradius of $\Omega$. We also obtain an upper bound and a lower bound of the blowup time. We use a comparison with suitable self-similar sub-solutions to prove the blowup and an upper bound of blowup time. Finally, we derive a lower bound of the blowup time by using the differential inequality technique.

Citation

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Yingying Liu. Zhengce Zhang. Liping Zhu. "Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption." Adv. Differential Equations 24 (3/4) 229 - 256, March/April 2019.

Information

Published: March/April 2019
First available in Project Euclid: 23 January 2019

zbMATH: 07192948
MathSciNet: MR3910034

Subjects:
Primary: 35B40, 35K15

Rights: Copyright © 2019 Khayyam Publishing, Inc.

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Vol.24 • No. 3/4 • March/April 2019
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