American Institute of Mathematical Sciences

Advanced
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 - 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

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

[1]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & 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 & Continuous Dynamical Systems - A, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[3]

Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & 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 & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[5]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225
[6]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317
[7]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[8]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
[9]

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
[10]

Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139
[11]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[12]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[13]

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
[14]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
[15]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[16]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[17]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[18]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523
[19]

Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691
[20]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (98)
  • HTML views (100)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences