## Differential and Integral Equations

### 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, $m=1$.

#### Article information

Source
Differential Integral Equations, Volume 9, Number 3 (1996), 447-464.

Dates
First available in Project Euclid: 7 May 2013

https://projecteuclid.org/euclid.die/1367969965

Mathematical Reviews number (MathSciNet)
MR1371701

Zentralblatt MATH identifier
0852.35063

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 80A20: Heat and mass transfer, heat flow

#### Citation

Chaves, Manuela; Vázquez, Juan Luis. Nonuniqueness in nonlinear heat propagation: a heat wave coming from infinity. Differential Integral Equations 9 (1996), no. 3, 447--464. https://projecteuclid.org/euclid.die/1367969965
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9844905734062195, "perplexity": 941.3172069130636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864172.45/warc/CC-MAIN-20180621133636-20180621153636-00172.warc.gz"}