Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Entire solutions and a Liouville theorem for a class of parabolic equations on the real line


Author: P. Poláčik
Journal: Proc. Amer. Math. Soc. 148 (2020), 2997-3008
MSC (2010): Primary 35K57, 35B40, 35B05
DOI: https://doi.org/10.1090/proc/14978
Published electronically: March 2, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a class of semilinear heat equations on $ \mathbb{R}$, including in particular the Fujita equation

$\displaystyle u_t=u_{xx} +\vert u\vert^{p-1}u,\quad x\in \mathbb{R},\ t\in \mathbb{R},$    

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35K57, 35B40, 35B05

Retrieve articles in all journals with MSC (2010): 35K57, 35B40, 35B05


Additional Information

P. Poláčik
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

DOI: https://doi.org/10.1090/proc/14978
Keywords: Semilinear parabolic equations, entire solutions, Liouville theorems, zero number
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
Article copyright: © Copyright 2020 American Mathematical Society