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

Poláčik, P.

doi:10.1090/proc/14978

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