A Liouville-type theorem for cooperative parabolic systems

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

February 2018, 38(2): 823-833. doi: 10.3934/dcds.2018035

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

Received December 2016 Revised September 2017 Published February 2018

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).

Keywords: Liouville-type theorem, parabolic system, singularity estimate, universal estimate, blow-up rate.

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

Citation: Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035

References:

[1]	H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.
[2]	T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.
[3]	J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory vol. 83 of Applied Mathematical Sciences, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-4546-9.
[4]	M. -F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, in Équations aux dérivées partielles et applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998,189–198.
[5]	M.-F. Bidaut-Véron and T. Raoux, Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations, 21 (1996), 1035-1086. doi: 10.1080/03605309608821217.
[6]	R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 533-559. doi: 10.1017/S0308210500025877.
[7]	E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769. doi: 10.1016/j.jde.2011.06.015.
[8]	E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.
[9]	M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202. doi: 10.1016/0022-0396(91)90118-S.
[10]	J. Földes and P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry, Discrete Contin. Dyn. Syst., 25 (2009), 133-157. doi: 10.3934/dcds.2009.25.133.
[11]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.
[12]	P. Glandsdorf and I. Prigogine, Thermodynamic Theory of Structure Stability and Fluctuations, 1971.
[13]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^N$, Comm. Partial Differential Equations, 33 (2008), 263-284. doi: 10.1080/03605300701257476.
[14]	H. Meinhardt, Models of Biological Pattern Formation vol. 6, Academic Press London, 1982.
[15]	Q. H. Phan, Optimal Liouville-type theorems for a parabolic system, Discrete Contin. Dyn. Syst., 35 (2015), 399-409. doi: 10.3934/dcds.2015.35.399.
[16]	Q. H. Phan and P. Souplet, A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems, Math. Ann., 366 (2016), 1561-1585. doi: 10.1007/s00208-016-1368-3.
[17]	P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Ⅱ. Parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908. doi: 10.1512/iumj.2007.56.2911.
[18]	P. Quittner, Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure, Math. Ann., 364 (2016), 269-292. doi: 10.1007/s00208-015-1219-7.
[19]	P. Quittner and P. Souplet, Superlinear Parabolic Problems Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007, Blow-up, global existence and steady states.
[20]	W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243. doi: 10.1006/jdeq.1999.3700.
[21]	J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.

show all references

References:

[1]	H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.
[2]	T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.
[3]	J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory vol. 83 of Applied Mathematical Sciences, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-4546-9.
[4]	M. -F. Bidaut-Véron, Initial blow-up for the solutions of a semilinear parabolic equation with source term, in Équations aux dérivées partielles et applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998,189–198.
[5]	M.-F. Bidaut-Véron and T. Raoux, Asymptotics of solutions of some nonlinear elliptic systems, Comm. Partial Differential Equations, 21 (1996), 1035-1086. doi: 10.1080/03605309608821217.
[6]	R. S. Cantrell, C. Cosner and V. Hutson, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 533-559. doi: 10.1017/S0308210500025877.
[7]	E. N. Dancer, K. Wang and Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J. Differential Equations, 251 (2011), 2737-2769. doi: 10.1016/j.jde.2011.06.015.
[8]	E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.
[9]	M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202. doi: 10.1016/0022-0396(91)90118-S.
[10]	J. Földes and P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry, Discrete Contin. Dyn. Syst., 25 (2009), 133-157. doi: 10.3934/dcds.2009.25.133.
[11]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.
[12]	P. Glandsdorf and I. Prigogine, Thermodynamic Theory of Structure Stability and Fluctuations, 1971.
[13]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^N$, Comm. Partial Differential Equations, 33 (2008), 263-284. doi: 10.1080/03605300701257476.
[14]	H. Meinhardt, Models of Biological Pattern Formation vol. 6, Academic Press London, 1982.
[15]	Q. H. Phan, Optimal Liouville-type theorems for a parabolic system, Discrete Contin. Dyn. Syst., 35 (2015), 399-409. doi: 10.3934/dcds.2015.35.399.
[16]	Q. H. Phan and P. Souplet, A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems, Math. Ann., 366 (2016), 1561-1585. doi: 10.1007/s00208-016-1368-3.
[17]	P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Ⅱ. Parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908. doi: 10.1512/iumj.2007.56.2911.
[18]	P. Quittner, Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure, Math. Ann., 364 (2016), 269-292. doi: 10.1007/s00208-015-1219-7.
[19]	P. Quittner and P. Souplet, Superlinear Parabolic Problems Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007, Blow-up, global existence and steady states.
[20]	W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243. doi: 10.1006/jdeq.1999.3700.
[21]	J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.

[1]	Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[2]	Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[3]	Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887
[4]	Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[5]	Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134
[6]	Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2073-2100. doi: 10.3934/dcds.2021184
[7]	Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[8]	Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317
[9]	Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[10]	Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
[11]	Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139
[12]	Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206
[13]	Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[14]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[15]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[16]	Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809
[17]	Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure and Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
[18]	Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic and Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[19]	Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[20]	C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure and Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Anh Tuan Duong Quoc Hung Phan
on Google Scholar
Anh Tuan Duong Quoc Hung Phan

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Anh Tuan Duong Quoc Hung Phan
on Google Scholar
Anh Tuan Duong Quoc Hung Phan

[Back to Top]