Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis
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| Format: | Preprint |
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2023
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| _version_ | 1866911856172466176 |
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| author | Muñoz-Hernández, Eduardo Sovrano, Elisa Taddei, Valentina |
| author_facet | Muñoz-Hernández, Eduardo Sovrano, Elisa Taddei, Valentina |
| contents | We investigate traveling wave solutions for a nonlinear system of two coupled reaction-diffusion equations characterized by double degenerate diffusivity: \[n_t= -f(n,b), \quad b_t=[g(n)h(b)b_x]_x+f(n,b).\] These systems mainly appear in modeling spatial-temporal patterns during bacterial growth. Central to our study is the diffusion term $g(n)h(b)$, which degenerates at $n=0$ and $b=0$; and the reaction term $f(n,b)$, which is positive, except for $n=0$ or $b=0$. Specifically, the existence of traveling wave solutions composed by a couple of strictly monotone functions for every wave speed in a closed half-line is proved, and some threshold speed estimates are given. Moreover, the regularity of the traveling wave solutions is discussed in connection with the wave speed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_05385 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis Muñoz-Hernández, Eduardo Sovrano, Elisa Taddei, Valentina Analysis of PDEs Classical Analysis and ODEs 35C07, 35K55, 35K57 We investigate traveling wave solutions for a nonlinear system of two coupled reaction-diffusion equations characterized by double degenerate diffusivity: \[n_t= -f(n,b), \quad b_t=[g(n)h(b)b_x]_x+f(n,b).\] These systems mainly appear in modeling spatial-temporal patterns during bacterial growth. Central to our study is the diffusion term $g(n)h(b)$, which degenerates at $n=0$ and $b=0$; and the reaction term $f(n,b)$, which is positive, except for $n=0$ or $b=0$. Specifically, the existence of traveling wave solutions composed by a couple of strictly monotone functions for every wave speed in a closed half-line is proved, and some threshold speed estimates are given. Moreover, the regularity of the traveling wave solutions is discussed in connection with the wave speed. |
| title | Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis |
| topic | Analysis of PDEs Classical Analysis and ODEs 35C07, 35K55, 35K57 |
| url | https://arxiv.org/abs/2311.05385 |