Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929474672525312 |
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| author | Columbu, Alessandro |
| author_facet | Columbu, Alessandro |
| contents | This work deals with the consumption chemotaxis problem
\begin{equation*}
\begin{cases*}
u_t = Δu - χ\nabla \cdot u\nabla v + λu - μu^2 - c \lvert \nabla u \rvert^γ, & \text{in $Ω\times(0,\tmax)$},
v_t = Δv - uv, & \text{in $Ω\times(0,\tmax)$},
\end{cases*} \end{equation*} in a bounded and smooth domain $Ω\subset\R^n$, $n\geq 3$, under Neumann boundary conditions, for $χ,λ,μ,c>0$, $\tmax\in(0,\infty]$ and for $u_0,v_0$ positive initial data with a certain regularity. We will show that the problem has a unique and uniformly bounded classical solution for $γ\in\bigl(\frac{2n}{n+1},2\bigr]$. Moreover, we have the same result for $γ=\frac{2n}{n+1}$ and a condition that involves the parameters $c,μ,n,χ$ and the initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities Columbu, Alessandro Analysis of PDEs This work deals with the consumption chemotaxis problem \begin{equation*} \begin{cases*} u_t = Δu - χ\nabla \cdot u\nabla v + λu - μu^2 - c \lvert \nabla u \rvert^γ, & \text{in $Ω\times(0,\tmax)$}, v_t = Δv - uv, & \text{in $Ω\times(0,\tmax)$}, \end{cases*} \end{equation*} in a bounded and smooth domain $Ω\subset\R^n$, $n\geq 3$, under Neumann boundary conditions, for $χ,λ,μ,c>0$, $\tmax\in(0,\infty]$ and for $u_0,v_0$ positive initial data with a certain regularity. We will show that the problem has a unique and uniformly bounded classical solution for $γ\in\bigl(\frac{2n}{n+1},2\bigr]$. Moreover, we have the same result for $γ=\frac{2n}{n+1}$ and a condition that involves the parameters $c,μ,n,χ$ and the initial data. |
| title | Boundedness criteria for a chemotaxis consumption model with gradient nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.14250 |