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

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## 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.## References

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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