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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Glandsdorf and I. Prigogine, Thermodynamic Theory of Structure Stability and Fluctuations, 1971. Google Scholar

[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.  Google Scholar

[14]

H. Meinhardt, Models of Biological Pattern Formation vol. 6, Academic Press London, 1982. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Glandsdorf and I. Prigogine, Thermodynamic Theory of Structure Stability and Fluctuations, 1971. Google Scholar

[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.  Google Scholar

[14]

H. Meinhardt, Models of Biological Pattern Formation vol. 6, Academic Press London, 1982. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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