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- W3170331393 abstract "This work deals with a parabolic chemotaxis model with nonlinear diffusion and nonlocal reaction source. The problem is formulated on the whole space and, depending on a specific interplay between the coefficients associated to such diffusion and reaction, we establish that all given solutions are uniformly bounded in time. To be precise, we study these attractive (sign ``$+$'') and repulsive (sign ``$-$'') following models, formally described by the Cauchy problems begin{equation}label{problem_abstract} tag{$Diamond$} begin{cases} displaystyle rho_t=Delta rho^m pm nabla cdot Big(rho nabla Big(frac{|x|^{2-n}}{2-n}*rhoBig)Big) [10pt] displaystyle hskip 50pt +arho^eta-brho^alpha int_{mathbb R^n} rho^beta dx & xin mathbb R^n, tin (0,T_{max}), rho(x,0)=rho_{0}(x) & xin mathbb R^n, end{cases} end{equation} for $ngeq 3$, $m,a,b,alpha,eta > 0$ and $betageq 1$. By denoting with $T_{max}$ the maximum time of existence of any nonnegative weak solution $rho$ to problems eqref{problem_abstract}, we prove that despite any large-mass initial data $rho_0$, for any $eta > 0$ and arbitrarily small diffusive parameter $m > 0$, whenever $alpha+beta$ surpasses some computable expression depending on $m, eta$ and $n$, $T_{max}=infty$ and $rho$ is uniformly bounded. On the one hand, this paper is in line with claims established for $a=b=0$, where the same conclusion holds true in, respectively: begin{align*} text{$bullet$} & text{the repulsive scenario, under the assumption $m > 0$}hskip 300pt & text{(adaptation of the case $m > 1-frac{2}{n}$, in Carrillo and Wang [17]);} text{$bullet$} & text{the attraction scenario, under the assumption $frac{2n}{n+2} < m < 2-frac{2}{n}$} & text{and for small initial data (Chen and Wang in [18]).} end{align*} On the other hand, for the attractive case with $a=b=m=1$ and $eta=alpha$, this investigation also extends a result derived by Bian, Chen and Latos in [3]." @default.
- W3170331393 created "2021-06-22" @default.
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- W3170331393 date "2021-04-01" @default.
- W3170331393 modified "2023-10-06" @default.
- W3170331393 title "Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime" @default.
- W3170331393 doi "https://doi.org/10.57262/die034-0506-315" @default.
- W3170331393 hasPublicationYear "2021" @default.
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