Nonuniqueness in nonlinear heat propagation: a heat wave coming from infinity

Abstract

It is well known that nonnegative solutions of the heat equation defined in a strip $Q=\mathbf{R}\times (0,T)$ are uniquely determined by their initial data. In this paper we construct a nonnegative solution $H(x,t)$ of the nonlinear heat equation $$ u_t=(u^m)_{xx}- u^p, \quad 1<p<m, \tag {E} $$ which takes on zero initial data, $H(x,0)\equiv 0$, and is nontrivial for $t\ge \tau>0$. This special solution has the following properties: (i) $H(x,t)$ is bounded for $t\ge \tau>0$. (ii) There exists a constant $r>0$ such that $$ \begin{align} & H(x,t)>0 \quad \rm {if }\quad rt^{-\beta} <x<\infty,\\ & H(x,t)=0 \quad \rm{if }\quad -\infty <x<rt^{-\beta},\end{align} $$ where $\beta=(m-p)/2(p-1)$. (iii) It has the self-similar form $H(x,t)= t^{-\alpha}f(xt^{\beta}),$ where $0\le f(\eta) \le c$ is supported in the interval $r\le \eta<\infty$. A similar phenomenon holds in several space dimensions with radial symmetry, where it describes a form of focussing. On the other hand, such a nonuniqueness does not occur for other ranges of $m$ and $p$. In particular, it does not happen for linear diffusion, $m=1$.