Nonuniqueness in nonlinear heat propagation: A heat wave coming from infinity (Q1909692)

Let \(m> p> 1\), \(\alpha:= 1/(p- 1)\) and \(\beta:= (m-p)/2(p- 1)\). The authors prove the existence of a unique number \(r> 0\) and a unique function \(f\in C^\infty([r, \infty))\) which is increasing and satisfies \[ (f^m)''(\eta)= \beta\eta f'(\eta)+ f^p(\eta),\quad \eta> 0, \] \[ f(r)= 0,\quad \lim_{n\downarrow r} f(\eta)^{m- 2} f'(\eta)= {r\beta\over m},\quad f(\infty)= (p- 1)^{-{1\over p-1}}. \] Extending \(f\) by \(0\) into \((- \infty, r)\) and setting \(H(t, x)= t^{- \alpha} f(t^\beta x)\) they obtain a weak solution \(H\) of \[ u_t= (u^m)_{xx}- u^p\tag{\(*\)} \] on \((0, \infty)\times \mathbb{R}\) which fulfills the initial condition \(u(0,\cdot)\equiv 0\) in the sense that \(H(t, x)> 0\), iff \(x> rt^{-\beta}\). They also classify all nonnegative self-similar solutions with bounded profiles of \((*)\). The introduction gives a useful discussion of earlier nonuniqueness results in this area.