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

Abstract

We consider a class of semilinear heat equations on R \mathbb {R} , including in particular the Fujita equation u t = u x x + | u | p − 1 u , x ∈ R , t ∈ R , \begin{equation*} u_t=u_{xx} +|u|^{p-1}u,\quad x\in \mathbb {R},\ t\in \mathbb {R}, \end{equation*} where p > 1 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.