Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916293244878848 |
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| author | Flüchter, Gregor |
| author_facet | Flüchter, Gregor |
| contents | We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity \begin{eqnarray*}
\left\{ \begin{array}{l}
u_t = \nabla \cdot (|x|^β\nabla u) - \nabla \cdot (u\nabla v), \\[1mm]
0 = Δv - μ+ u,
\qquad μ:=\frac{1}{|Ω|} \int\limits_Ωu,
\end{array} \right.
\qquad \qquad (\star)
\end{eqnarray*} under homogeneous Neumann boundary conditions in the ball $Ω=B_R(0)\subset \mathbb R^n$.
For $β>0$ and radially symmetric Hölder continuous initial data, we prove that there exists a pointwise classical solution to $(\star)$ in $(Ω\setminus \{0\})\times (0,T)$ for some $T>0$. For radially decreasing initial data satisfying certain compatibility criteria, this solution is bounded and unique in $(Ω\setminus \{0\})\times (0,T^*)$ for some $T^*>0$. Moreover, for $n \geq 2$ and sufficiently accumulated initial data, there exists no solution $(u,v)$ to $(\star)$ in the sense specified above which is globally bounded in time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity Flüchter, Gregor Analysis of PDEs 35A01, 35K65 (primary), 35A02, 35B44, 35B33, 92C17 (secondary) We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (|x|^β\nabla u) - \nabla \cdot (u\nabla v), \\[1mm] 0 = Δv - μ+ u, \qquad μ:=\frac{1}{|Ω|} \int\limits_Ωu, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} under homogeneous Neumann boundary conditions in the ball $Ω=B_R(0)\subset \mathbb R^n$. For $β>0$ and radially symmetric Hölder continuous initial data, we prove that there exists a pointwise classical solution to $(\star)$ in $(Ω\setminus \{0\})\times (0,T)$ for some $T>0$. For radially decreasing initial data satisfying certain compatibility criteria, this solution is bounded and unique in $(Ω\setminus \{0\})\times (0,T^*)$ for some $T^*>0$. Moreover, for $n \geq 2$ and sufficiently accumulated initial data, there exists no solution $(u,v)$ to $(\star)$ in the sense specified above which is globally bounded in time. |
| title | Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity |
| topic | Analysis of PDEs 35A01, 35K65 (primary), 35A02, 35B44, 35B33, 92C17 (secondary) |
| url | https://arxiv.org/abs/2406.12633 |