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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.02741 |
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| _version_ | 1866909304777342976 |
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| author | Wu, Duan |
| author_facet | Wu, Duan |
| contents | This work considers the doubly degenerate nutrient model \begin{equation*}\label{AH1} \left\{ \begin{split} &u_t=\nabla\cdot\left(u^{m-1}v\nabla u\right)-\nabla\cdot\left(f(u)v\nabla v\right)+\ell uv,&&x\inΩ,\,t>0, &v_t=Δv-uv, &&x\inΩ,\,t>0, \end{split} \right. \end{equation*} under no-flux boundary conditions in a smoothly bounded convex domain $Ω\subset \mathbb{R}^n$ ($n\le 2$), where the nonnegative function $f\in C^1([0,\infty))$ is assumed to satisfy $f(s)\le C_fs^α$ with $α>0$ and $C_f>0$ for all $s\ge 1$. When $m=2$, it was shown that a global weak solution exists, either in one-dimensional setting with $α=2$, or in two-dimensional version with $α\in(1,\frac{3}{2})$. The main results in this paper assert the global existence of weak solutions for $1\le m<3$ and classical solutions for $3\le m<4$ to the above system under the assumption \begin{equation*} α\in \left\{ \begin{split} &\left[m-1,\min\left\{m,\frac{m}{2}+1\right\}\right]~~&&\textrm{if}~~n=1,\quad\quad\textrm{and} &\left(m-1,\min\left\{m,\frac{m}{2}+1\right\}\right)~~&&\textrm{if}~~n=2, \end{split} \right. \end{equation*} which extend the range $α\in(1,\frac{3}{2})$ to $α\in(1,2)$ in two dimensions for the case $m=2$. Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Refined existence theorems for doubly degenerate chemotaxis-consumption systems with large initial data Wu, Duan Analysis of PDEs This work considers the doubly degenerate nutrient model \begin{equation*}\label{AH1} \left\{ \begin{split} &u_t=\nabla\cdot\left(u^{m-1}v\nabla u\right)-\nabla\cdot\left(f(u)v\nabla v\right)+\ell uv,&&x\inΩ,\,t>0, &v_t=Δv-uv, &&x\inΩ,\,t>0, \end{split} \right. \end{equation*} under no-flux boundary conditions in a smoothly bounded convex domain $Ω\subset \mathbb{R}^n$ ($n\le 2$), where the nonnegative function $f\in C^1([0,\infty))$ is assumed to satisfy $f(s)\le C_fs^α$ with $α>0$ and $C_f>0$ for all $s\ge 1$. When $m=2$, it was shown that a global weak solution exists, either in one-dimensional setting with $α=2$, or in two-dimensional version with $α\in(1,\frac{3}{2})$. The main results in this paper assert the global existence of weak solutions for $1\le m<3$ and classical solutions for $3\le m<4$ to the above system under the assumption \begin{equation*} α\in \left\{ \begin{split} &\left[m-1,\min\left\{m,\frac{m}{2}+1\right\}\right]~~&&\textrm{if}~~n=1,\quad\quad\textrm{and} &\left(m-1,\min\left\{m,\frac{m}{2}+1\right\}\right)~~&&\textrm{if}~~n=2, \end{split} \right. \end{equation*} which extend the range $α\in(1,\frac{3}{2})$ to $α\in(1,2)$ in two dimensions for the case $m=2$. Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form. |
| title | Refined existence theorems for doubly degenerate chemotaxis-consumption systems with large initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.02741 |