A triviality result for semilinear parabolic equations

Daniele Castorina; Giovanni Catino; Carlo Mantegazza; Daniele Castorina; Giovanni Catino; Carlo Mantegazza

doi:10.3934/mine.2022002

Mathematics in Engineering

2022, Volume 4, Issue 1: 1-15. doi: 10.3934/mine.2022002

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A triviality result for semilinear parabolic equations

1.
Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Cintia, Monte S. Angelo 80126 Napoli, Italy
2.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

Received: 15 July 2020 Accepted: 30 January 2021 Published: 10 March 2021

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation

$ \begin{equation*} u_{t} = \Delta u + |u|^{p} \end{equation*} $

on complete Riemannian manifolds of dimension $ n \geq 5 $ with nonnegative Ricci tensor, when $ p $ is smaller than the critical Sobolev exponent $ \frac{n+2}{n-2} $.
- ancient solution,
- eternal solution,
- superlinear heat equation
Citation: Daniele Castorina, Giovanni Catino, Carlo Mantegazza. A triviality result for semilinear parabolic equations[J]. Mathematics in Engineering, 2022, 4(1): 1-15. doi: 10.3934/mine.2022002

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Abstract

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation

$ \begin{equation*} u_{t} = \Delta u + |u|^{p} \end{equation*} $

on complete Riemannian manifolds of dimension $ n \geq 5 $ with nonnegative Ricci tensor, when $ p $ is smaller than the critical Sobolev exponent $ \frac{n+2}{n-2} $.

References

[1]	D. G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291
[2]	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, Paris: Gauthier-Villars, 1998,189-198.
[3]	M. F. Bidaut-Véron, T. Raoux, Asymptotics of solutions of some nonlinear elliptic systems, Commun. Part. Diff. Eq., 21 (1996), 1035-1086. doi: 10.1080/03605309608821217
[4]	D. Castorina, C. Mantegazza, Ancient solutions of semilinear heat equations on Riemannian manifolds, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 85-101. doi: 10.4171/RLM/753
[5]	D. Castorina, C. Mantegazza, Ancient solutions of superlinear heat equations on Riemannian manifolds, Commun. Contemp. Math., 23 (2021), 2050033. doi: 10.1142/S0219199720500339
[6]	S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Springer-Verlag, 1990.
[7]	C. Gui, W. M. Ni, X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in ${\bf{R}}^n$, Commun. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906
[8]	B. Güneysu, Sequences of Laplacian cut-off functions, J. Geom. Anal., 26 (2016), 171-184. doi: 10.1007/s12220-014-9543-9
[9]	P. Petersen, Riemannian geometry, 2 Eds., New York: Springer, 2006.
[10]	P. Poláčik, P. Quittner, 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
[11]	P. Poláčik, P. Quittner, 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
[12]	P. Quittner, Optimal Liouville theorems for superlinear parabolic problems, arXiv: 2003.13223.
[13]	L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., 36 (1992), 417-450. doi: 10.4310/jdg/1214448748
[14]	G. Talenti, Best constant in Sobolev inequality, Annali di Matematica, 110 (1976), 353-372. doi: 10.1007/BF02418013

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