For \(p>2\) , the equation \(\begin{aligned} u_t = u^p u_{xx}, \qquad x\in \mathbb {R}, \ t\in \mathbb {R}, \end{aligned}\) is shown to admit positive and spatially increasing smooth solutions on all of \(\mathbb {R}\times \mathbb {R}\) which are precisely of the form of an accelerating wave for \(t<0\) , and of a wave slowing down for \(t>0\) . These solutions satisfy \(u(\cdot ,t)\rightarrow 0\) in \(L^\infty _{loc}(\mathbb {R})\) as \(t\rightarrow + \infty \) and as \(t\rightarrow -\infty \) , and exhibit a yet apparently undiscovered phenomenon of transient rapid spatial growth, in the sense that \(\begin{aligned} \lim _{x\rightarrow +\infty } x^{-1} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t<0, \end{aligned}\) that \(\begin{aligned} \lim _{x\rightarrow +\infty } x^{-\frac{2}{p}} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t>0, \end{aligned}\) but that \(\begin{aligned} u(x,0)=K e^{\alpha x} \qquad \text{ for } \text{ all } x\in \mathbb {R}\end{aligned}\) with some \(K>0\) and \(\alpha >0\) .
