Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime

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}*\rho\Big)\Big) \\[10pt] \displaystyle \hskip 50pt +a\rho^\eta-b\rho^\alpha \int_{\mathbb R^n} \rho^\beta dx & x\in \mathbb R^n, t\in (0,T_{\max}), \\ \rho(x,0)=\rho_{0}(x) & x\in \mathbb R^n, \end{cases} \end{equation} for $n\geq 3$, $m,a,b,\alpha,\eta > 0$ and $\beta\geq 1$. 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].