January  2015, 35(1): 399-409. doi: 10.3934/dcds.2015.35.399

Optimal Liouville-type theorems for a parabolic system

1. 

Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

Received  September 2013 Revised  June 2014 Published  August 2014

We prove Liouville-type theorems for a parabolic system in dimension $N=1$ and for radial solutions in all dimensions under an optimal Sobolev growth restriction on the nonlinearities. This seems to be the first example of a Liouville-type theorem in the whole Sobolev subcritical range for a parabolic system (even for radial solutions). Moreover, this also seems to be the first application of the Gidas-Spruck technique to a parabolic system.
Citation: 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
References:
[1]

K. Ammar and P. Souplet, Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source,, Discrete Contin. Dyn. Syst., 26 (2010), 665. doi: 10.3934/dcds.2010.26.665.

[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. doi: 10.1007/s00526-009-0265-y.

[3]

T. Bartsch, P. Poláčik and P. Quittner, Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations,, J. Eur. Math. Soc. (JEMS), 13 (2011), 219. doi: 10.4171/JEMS/250.

[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, (1998), 189.

[5]

M.-F. Bidaut-Véron and T. Raoux, Asymptotics of solutions of some nonlinear elliptic systems,, Comm. Partial Differential Equations, 21 (1996), 1035. doi: 10.1080/03605309608821217.

[6]

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. doi: 10.1016/j.anihpc.2010.01.009.

[7]

E. N. Dancer and T. Weth, Liouville-type results for non-cooperative elliptic systems in a half-space,, J. Lond. Math. Soc. (2), 86 (2012), 111. doi: 10.1112/jlms/jdr080.

[8]

Y. Guo, B. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents,, J. Diff. Equations, 256 (2014), 3463. doi: 10.1016/j.jde.2014.02.007.

[9]

P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, Duke Math. J., 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[10]

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. doi: 10.1512/iumj.2007.56.2911.

[11]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], (2007).

[12]

P. Quittner and P. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts,, Discrete Contin. Dyn. Syst., (2011), 1206. doi: 10.3934/proc.2011.2011.1206.

[13]

P. Quittner and P. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications,, Comm. Math. Phys., 311 (2012), 1. doi: 10.1007/s00220-012-1440-0.

[14]

H. Tavares, S. Terracini, G. Verzini and T. Weth, Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems,, Comm. Partial Differential Equations, 36 (2011), 1988. doi: 10.1080/03605302.2011.574244.

[15]

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. doi: 10.1007/s00205-008-0121-9.

show all references

References:
[1]

K. Ammar and P. Souplet, Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source,, Discrete Contin. Dyn. Syst., 26 (2010), 665. doi: 10.3934/dcds.2010.26.665.

[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. doi: 10.1007/s00526-009-0265-y.

[3]

T. Bartsch, P. Poláčik and P. Quittner, Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations,, J. Eur. Math. Soc. (JEMS), 13 (2011), 219. doi: 10.4171/JEMS/250.

[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, (1998), 189.

[5]

M.-F. Bidaut-Véron and T. Raoux, Asymptotics of solutions of some nonlinear elliptic systems,, Comm. Partial Differential Equations, 21 (1996), 1035. doi: 10.1080/03605309608821217.

[6]

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. doi: 10.1016/j.anihpc.2010.01.009.

[7]

E. N. Dancer and T. Weth, Liouville-type results for non-cooperative elliptic systems in a half-space,, J. Lond. Math. Soc. (2), 86 (2012), 111. doi: 10.1112/jlms/jdr080.

[8]

Y. Guo, B. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents,, J. Diff. Equations, 256 (2014), 3463. doi: 10.1016/j.jde.2014.02.007.

[9]

P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems,, Duke Math. J., 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[10]

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. doi: 10.1512/iumj.2007.56.2911.

[11]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], (2007).

[12]

P. Quittner and P. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts,, Discrete Contin. Dyn. Syst., (2011), 1206. doi: 10.3934/proc.2011.2011.1206.

[13]

P. Quittner and P. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications,, Comm. Math. Phys., 311 (2012), 1. doi: 10.1007/s00220-012-1440-0.

[14]

H. Tavares, S. Terracini, G. Verzini and T. Weth, Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems,, Comm. Partial Differential Equations, 36 (2011), 1988. doi: 10.1080/03605302.2011.574244.

[15]

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. doi: 10.1007/s00205-008-0121-9.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[10]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[11]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605

[12]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723

[13]

Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469

[14]

Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243

[15]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[16]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091

[17]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[18]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020

[19]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129

[20]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]