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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Entire solutions and a Liouville theorem for a class of parabolic equations on the real line
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by P. Poláčik PDF
Proc. Amer. Math. Soc. 148 (2020), 2997-3008 Request permission

Abstract:

We consider a class of semilinear heat equations on $\mathbb {R}$, including in particular the Fujita equation \begin{equation*} u_t=u_{xx} +|u|^{p-1}u,\quad x\in \mathbb {R},\ t\in \mathbb {R}, \end{equation*} where $p>1$. We first give a simple proof and an extension of a Liouville theorem concerning entire solutions with finite zero number. Then we show that there is an infinite-dimensional set of entire solutions with infinite zero number.
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Additional Information
  • P. Poláčik
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • Received by editor(s): June 18, 2018
  • Received by editor(s) in revised form: November 25, 2019
  • Published electronically: March 2, 2020
  • Additional Notes: This research was supported in part by NSF Grant DMS-1565388
  • Communicated by: Ryan Hynd
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 2997-3008
  • MSC (2010): Primary 35K57, 35B40, 35B05
  • DOI: https://doi.org/10.1090/proc/14978
  • MathSciNet review: 4099786