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November  2021, 29(5): 2829-2839. doi: 10.3934/era.2021016

A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart

Christos Sourdis

National and Kapodistrian University of Athens, Department of Mathematics, Athens, Greece

Received  August 2020 Revised  December 2020 Published  November 2021 Early access  March 2021

Fund Project: This work has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No 1889

We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $ u_t = \Delta u+|u|^{p-1}u $ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $ p $ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.

Keywords: Semilinear heat equation, ancient solutions, semilinear elliptic equation, entire solutions, stable solutions, Liouville theorems, maximum principle.
Mathematics Subject Classification: Primary: 35B08, 35B53; Secondary: 35B40.
Citation: Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, 2021, 29 (5) : 2829-2839. doi: 10.3934/era.2021016
References:
[1]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar

[2]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal., 9 (1999), 221-246.  doi: 10.1007/BF02921937.  Google Scholar

[4] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 143, CRC Press, Boca Raton, 2011.  doi: 10.1201/b10802.  Google Scholar
[5]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbouned domains of $\mathbb{R}^N$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[6]

M. Fila and E. Yanagida, Homoclinic and heteroclinic orbits for a semilinear parabolic equation, Tohoku Math. J., 63 (2011), 561-579.  doi: 10.2748/tmj/1325886281.  Google Scholar

[7]

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

[8]

C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[9]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[10]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[11]

A. McNabb, Strong comparison theorems for elliptic equations of second order, J. Math. Mech., 10 (1961), 431-440.   Google Scholar

[12]

Y. Naito, An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 807-835.  doi: 10.1017/S0308210500004741.  Google Scholar

[13]

P. Poláčik and P. Quittner, Entire and ancient solutions of a supercritical semilinear heat equation, Discrete Cont. Dynamical Syst., 41 (2021), 413-438.  doi: 10.3934/dcds.2020136.  Google Scholar

[14]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[15]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908.  doi: 10.1512/iumj.2007.56.2911.  Google Scholar

[16]

P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214.  doi: 10.1016/j.jde.2003.10.019.  Google Scholar

[17]

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

[18]

P. Quittner, Optimal Liouville theorems for superlinear parabolic problems, Duke Math. J., to appear. Google Scholar

[19]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, 2$^nd$ edition, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2019. doi: 10.1007/978-3-030-18222-9.  Google Scholar

[20]

R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Topics in Calculus of Variations, Lecture Notes in Math. Springer, 1365 (1989), 120–154. doi: 10.1007/BFb0089180.  Google Scholar

[21]

C. Sourdis, A Liouville property for eternal solutions to a supercritical semilinear heat equation, preprint, arXiv: 1909.00498. Google Scholar

show all references

References:
[1]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar

[2]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal., 9 (1999), 221-246.  doi: 10.1007/BF02921937.  Google Scholar

[4] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 143, CRC Press, Boca Raton, 2011.  doi: 10.1201/b10802.  Google Scholar
[5]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbouned domains of $\mathbb{R}^N$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[6]

M. Fila and E. Yanagida, Homoclinic and heteroclinic orbits for a semilinear parabolic equation, Tohoku Math. J., 63 (2011), 561-579.  doi: 10.2748/tmj/1325886281.  Google Scholar

[7]

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

[8]

C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar

[9]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508.  Google Scholar

[10]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[11]

A. McNabb, Strong comparison theorems for elliptic equations of second order, J. Math. Mech., 10 (1961), 431-440.   Google Scholar

[12]

Y. Naito, An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 807-835.  doi: 10.1017/S0308210500004741.  Google Scholar

[13]

P. Poláčik and P. Quittner, Entire and ancient solutions of a supercritical semilinear heat equation, Discrete Cont. Dynamical Syst., 41 (2021), 413-438.  doi: 10.3934/dcds.2020136.  Google Scholar

[14]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[15]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations, Indiana Univ. Math. J., 56 (2007), 879-908.  doi: 10.1512/iumj.2007.56.2911.  Google Scholar

[16]

P. Poláčik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214.  doi: 10.1016/j.jde.2003.10.019.  Google Scholar

[17]

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

[18]

P. Quittner, Optimal Liouville theorems for superlinear parabolic problems, Duke Math. J., to appear. Google Scholar

[19]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, 2$^nd$ edition, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2019. doi: 10.1007/978-3-030-18222-9.  Google Scholar

[20]

R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Topics in Calculus of Variations, Lecture Notes in Math. Springer, 1365 (1989), 120–154. doi: 10.1007/BFb0089180.  Google Scholar

[21]

C. Sourdis, A Liouville property for eternal solutions to a supercritical semilinear heat equation, preprint, arXiv: 1909.00498. Google Scholar

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