Advances in Differential Equations

Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption

Yingying Liu, Zhengce Zhang, and Liping Zhu

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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.

Article information

Adv. Differential Equations, Volume 24, Number 3/4 (2019), 229-256.

First available in Project Euclid: 23 January 2019

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 35K15: Initial value problems for second-order parabolic equations 35B40: Asymptotic behavior of solutions


Liu, Yingying; Zhang, Zhengce; Zhu, Liping. Global existence and blowup for a quasilinear parabolic equations with nonlinear gradient absorption. Adv. Differential Equations 24 (2019), no. 3/4, 229--256.

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