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1996 Nonuniqueness in nonlinear heat propagation: a heat wave coming from infinity
Manuela Chaves, Juan Luis Vázquez
Differential Integral Equations 9(3): 447-464 (1996). DOI: 10.57262/die/1367969965


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, $m=1$.


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Manuela Chaves. Juan Luis Vázquez. "Nonuniqueness in nonlinear heat propagation: a heat wave coming from infinity." Differential Integral Equations 9 (3) 447 - 464, 1996.


Published: 1996
First available in Project Euclid: 7 May 2013

zbMATH: 0852.35063
MathSciNet: MR1371701
Digital Object Identifier: 10.57262/die/1367969965

Primary: 35K55
Secondary: 80A20

Rights: Copyright © 1996 Khayyam Publishing, Inc.


Vol.9 • No. 3 • 1996
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