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- Title
Dynamics in a parabolic-elliptic chemotaxis system with logistic source involving exponents depending on the spatial variables.
- Authors
Ayazoglu, Rabil; Kadakal, Mahir; Akkoyunlu, Ebubekir
- Abstract
We consider the parabolic-elliptic chemotaxis system with the exponents depending on the spatial variableslogistic source and nonlinear signal production: $ u_{t} = \Delta u-\chi \nabla \cdot \left(u\nabla \upsilon \right) +f(x,u),(x,t)\in \Omega \times (0,T) $, $ 0 = \Delta \upsilon -\upsilon +u^{\gamma } $ in a bounded domain $ \Omega \subset \mathbb{R} ^{N} $ $ \left(N\geq 1\right) $ with smooth boundary, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where $ \chi >0 $, $ \gamma \geq 1 $ and $ \frac{\partial }{\partial \nu } $ denotes the outward normal derivative on $ \partial \Omega $. The logistic function $ f $ fulfilling $ f(x,s)\leq \eta s-\mu s^{\alpha \left(x\right) +1} $, $ \eta \geq 0 $, $ \mu >0 $ for all $ s>0 $ with $ f(x,0)\geq 0 $ $ \forall x\in \Omega $, where $ \alpha :\Omega \rightarrow \left[ 1,\infty \right) $ is a measurable function. It is proved that if $ 1\leq \alpha \left(x\right) <\infty $ for all $ x\in \Omega $ such that $ ess\inf_{x\in \Omega }\alpha \left(x\right) >\gamma $ or$ ess\inf_{x\in \Omega }\alpha \left(x\right) = \gamma $ with $ \mu >\chi $, then there exists a nonnegative classical solution $ (u,\upsilon) $ that is global-in-time and bounded. In addition, under the particular conditions $ \gamma = 1 $ and $ f(x,s) = \mu \left(s-s^{\alpha \left(x\right) +1}\right) $, if $ \mu $ is sufficiently large, the global bounded solution $ (u,\upsilon) $ satisfies$ \begin{equation*} \left\Vert u\left(\cdot ,t\right) -1\right\Vert _{L^{\infty }\left(\Omega \right) }+\left\Vert \upsilon \left(\cdot ,t\right) -1\right\Vert _{L^{\infty }\left(\Omega \right) }\leq Ce^{-\frac{k}{N+2}t}\text{ } \end{equation*} $for all$ \ t>0 $ with $ k = \min \left\{ \frac{\chi ^{2}}{4},\frac{1}{2}\right\} $, $ C>0 $.The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameter conditions, which improves the known results.
- Subjects
MATHEMATICAL logic; CHEMOTAXIS; NEUMANN boundary conditions; EXPONENTS; LYAPUNOV exponents; PARABOLIC operators
- Publication
Discrete & Continuous Dynamical Systems - Series B, 2024, Vol 29, Issue 5, p1
- ISSN
1531-3492
- Publication type
Article
- DOI
10.3934/dcdsb.2023169