# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 665-689. 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), 345-361. 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), 219-247. 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, 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] 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. [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), 111-128. 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), 3463-3495. 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), 555-579. 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), 879-908. 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], Birkhäuser Verlag, Basel, 2007. [12] P. Quittner and P. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts, Discrete Contin. Dyn. Syst., suppl. (2011), 1206-1213. 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), 1-19. 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), 1988-2010. 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), 83-106. doi: 10.1007/s00205-008-0121-9.

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##### 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), 665-689. 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), 345-361. 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), 219-247. 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, 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] 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. [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), 111-128. 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), 3463-3495. 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), 555-579. 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), 879-908. 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], Birkhäuser Verlag, Basel, 2007. [12] P. Quittner and P. Souplet, Parabolic Liouville-type theorems via their elliptic counterparts, Discrete Contin. Dyn. Syst., suppl. (2011), 1206-1213. 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), 1-19. 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), 1988-2010. 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), 83-106. doi: 10.1007/s00205-008-0121-9.
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