A Liouville-type theorem for cooperative parabolic systems

Anh Tuan Duong; Quoc Hung Phan

doi:10.3934/dcds.2018035

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2018, Volume 38, Issue 2: 823-833. Doi: 10.3934/dcds.2018035

This issue Previous Article N-barrier maximum principle for degenerate elliptic systems and its application Next Article Reversing and extended symmetries of shift spaces

A Liouville-type theorem for cooperative parabolic systems

Anh Tuan Duong^1, and
Quoc Hung Phan^2, ,

1.
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Hanoi, Vietnam
2.
Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

* Corresponding author: Quoc Hung Phan
* Corresponding author: Quoc Hung Phan

Received: November 30, 2016

Revised: August 31, 2017

Published: November 2017

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Abstract

We prove Liouville-type theorem for semilinear parabolic system of the form $u_t-\Delta u =a_{11}u^{p}+a_{12} u^rv^{s+1}$, $v_t-\Delta v =a_{21} u^{r+1}v^{s}+a_{22}v^{p}$ where $r, s>0$, $p=r+s+1$. The real matrix $A=(a_{ij})$ satisfies conditions $ a_{12}, a_{21}\geq 0$ and $a_{11}, a_{22}>0$. This paper is a continuation of Phan-Souplet (Math. Ann., 366,1561-1585,2016) where the authors considered the special case $s=r$ for the system of $m$ components. Our tool for the proof of Liouville-type theorem is a refinement of Phan-Souplet, which is based on Gidas-Spruck (Commun. Pure Appl.Math. 34,525–598 1981) and Bidaut-Véron (Équations aux dérivées partielles et applications. Elsevier, Paris, pp 189–198,1998).

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Mathematics Subject Classification: Primary: 35B53, 35K58; Secondary: 35B33, 35B44, 35B45.

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