Global boundedness of solutions to a parabolic–parabolic chemotaxis system with local sensing in higher dimensions
نویسندگان
چکیده
This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=\Delta (\gamma (v) u ) &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] v_t=\Delta v - + \displaystyle \frac{\partial u}{\partial \nu} = v}{\partial 0 &\mathrm{on}\ \partial\Omega \times (0,\infty), u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 \Omega, \end{cases} \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^n$($n \geq 3$), $\gamma (v)=v^{-k}$ ($k>0$) and initial data $(u_0,v_0)$ positive regular. has striking features similar those of logarithmic Keller--Segel system. It established that exist globally time remain uniformly if $k \in (0,n/(n-2))$, independently magnitude mass. constant $n/(n-2)$ conjectured as optimal range guaranteeing global existence boundedness corresponding We will derive sufficient estimates for through some single evolution equation auxiliary function satisfies. The cornerstone analysis refined comparison estimate solutions, which enables us control nonlinearity equation.
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ژورنال
عنوان ژورنال: Nonlinearity
سال: 2022
ISSN: ['0951-7715', '1361-6544']
DOI: https://doi.org/10.1088/1361-6544/ac6659