# American Institute of Mathematical Sciences

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

## A Liouville-type theorem for cooperative parabolic systems

 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).

Citation: Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems - A, 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

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##### 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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