Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation

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 $$\alpha >0$$ .