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

Christos Sourdis; Christos Sourdis

doi:10.3934/era.2021016

Electronic Research Archive

2021, Volume 29, Issue 5: 2829-2839. doi: 10.3934/era.2021016

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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: 01 August 2020 Revised: 01 December 2020 Published: 15 March 2021
Primary: 35B08, 35B53; Secondary: 35B40

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.
- Semilinear heat equation,
- ancient solutions,
- semilinear elliptic equation,
- entire solutions,
- stable solutions,
- Liouville theorems,
- maximum principle
Citation: Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart[J]. Electronic Research Archive, 2021, 29(5): 2829-2839. doi: 10.3934/era.2021016

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Abstract

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.

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