Electron. J. Diff. Equ., Vol. 2016 (2016), No. 45, pp. 1-5.

Non-extinction of solutions to a fast diffusion system with nonlocal sources

Haixia Li, Yuzhu Han

Abstract:
In this short article, we give a positive answer to the problem proposed by Zheng et al [5], and show that the fast diffusion system
$$\displaylines{ u_t=\hbox{div}(|\nabla u|^{p-2}\nabla u) +\int_\Omega v^\alpha\,dx, \cr v_t =\hbox{div}(|\nabla v|^{q-2}\nabla v) +\int_\Omega u^\beta\, dx }$$
under homogeneous Dirichlet boundary condition admits at least one non-extinction solution when $\alpha\beta<(p-1)(q-1)$ and the initial data are strictly positive.

Submitted October 28, 2015. Published February 10, 2016.
Math Subject Classifications: 35K40, 35K51.
Key Words: Fast diffusion system; nonlocal source; non-extinction.

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Haixia Li
School of Mathematics
Changchun Normal University
Changchun 130032, China
email: lihaixia0611@126.com
Yuzhu Han
School of Mathematics
Jilin University
Changchun 130012, China
email: yzhan@jlu.edu.cn

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