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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2409.12083 |
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| _version_ | 1866911228557787136 |
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| author | Wu, Duan |
| author_facet | Wu, Duan |
| contents | The present work proceeds to consider the convergence of the solutions to the following doubly degenerate chemotaxis-consumption system \begin{align*} \left\{ \begin{array}{r@{\,}l@{\quad}l@{\,}c} &u_{t}=\nabla\cdot\big(u^{m-1}v\nabla v\big)-\nabla\cdot\big(f(u)v\nabla v\big)+\ell uv,\\ &v_{t}=Δv-uv, \end{array}\right.%} \end{align*} under no-flux boundary conditions in a smoothly bounded convex domain $Ω\subset \R^2$, where the nonnegative function $f\in C^1([0,\infty))$ is asked to satisfy $f(s)\le C_fs^{\al}$ with $\al, C_f>0$ for all $s\ge 1$.
The global existence of weak solutions or classical solutions to the above system has been established in both one- and two-dimensional bounded convex domains in previous works. However, the results concerning the large time behavior are still constrained to one dimension due to the lack of a Harnack-type inequality in the two-dimensional case. In this note, we complement this result by using the Moser iteration technique and building a new Harnack-type inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The asymptotic behavior of solutions to a doubly degenerate chemotaxis-consumption system in the two-dimensional setting Wu, Duan Analysis of PDEs The present work proceeds to consider the convergence of the solutions to the following doubly degenerate chemotaxis-consumption system \begin{align*} \left\{ \begin{array}{r@{\,}l@{\quad}l@{\,}c} &u_{t}=\nabla\cdot\big(u^{m-1}v\nabla v\big)-\nabla\cdot\big(f(u)v\nabla v\big)+\ell uv,\\ &v_{t}=Δv-uv, \end{array}\right.%} \end{align*} under no-flux boundary conditions in a smoothly bounded convex domain $Ω\subset \R^2$, where the nonnegative function $f\in C^1([0,\infty))$ is asked to satisfy $f(s)\le C_fs^{\al}$ with $\al, C_f>0$ for all $s\ge 1$. The global existence of weak solutions or classical solutions to the above system has been established in both one- and two-dimensional bounded convex domains in previous works. However, the results concerning the large time behavior are still constrained to one dimension due to the lack of a Harnack-type inequality in the two-dimensional case. In this note, we complement this result by using the Moser iteration technique and building a new Harnack-type inequality. |
| title | The asymptotic behavior of solutions to a doubly degenerate chemotaxis-consumption system in the two-dimensional setting |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.12083 |