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Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations
A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart
National and Kapodistrian University of Athens, Department of Mathematics, Athens, Greece |
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.
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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[11] |
A. McNabb,
Strong comparison theorems for elliptic equations of second order, J. Math. Mech., 10 (1961), 431-440.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[11] |
A. McNabb,
Strong comparison theorems for elliptic equations of second order, J. Math. Mech., 10 (1961), 431-440.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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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