Refined estimates of the propagation speed in porous medium equation of combustion type
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909040117809152 |
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| author | Liu, Suying Wu, Fan |
| author_facet | Liu, Suying Wu, Fan |
| contents | We are concerned with the Cauchy problem $u_{t}=(u^{m})_{xx}+f(u)$, where the nonliearity $f(u)$ is of combustion type and the initial data is compactly supported. In \cite{lou2024convergence}, among other things, the authors prove that by considering a multiple of a given initial data, there is a critical value such that the corresponding transition solution spreads at the asymptotic speed $2y_{0}\sqrt{t}[1+o(1)]\ \text{as} \ t\rightarrow\infty$, while the lower order term $o(1)$ remains unknown. In this paper, for a family of functions of combustion type, we refine the estimates of the asymptotic speed of the transition solution and provide a precise characterization of the lower order term $o(1)$. Our result also reveals that there is no unified characterization of the lower order term for general combustion type functions $f$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Refined estimates of the propagation speed in porous medium equation of combustion type Liu, Suying Wu, Fan Analysis of PDEs We are concerned with the Cauchy problem $u_{t}=(u^{m})_{xx}+f(u)$, where the nonliearity $f(u)$ is of combustion type and the initial data is compactly supported. In \cite{lou2024convergence}, among other things, the authors prove that by considering a multiple of a given initial data, there is a critical value such that the corresponding transition solution spreads at the asymptotic speed $2y_{0}\sqrt{t}[1+o(1)]\ \text{as} \ t\rightarrow\infty$, while the lower order term $o(1)$ remains unknown. In this paper, for a family of functions of combustion type, we refine the estimates of the asymptotic speed of the transition solution and provide a precise characterization of the lower order term $o(1)$. Our result also reveals that there is no unified characterization of the lower order term for general combustion type functions $f$. |
| title | Refined estimates of the propagation speed in porous medium equation of combustion type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.13361 |