Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction–repulsion chemotaxis models with logistics

Abstract

The following fully nonlinear attraction–repulsion and zero-flux chemotaxis model is studied: ($♢$)$\left(\begin{array}{c}{u}_t=\nabla \cdot \left({(u+1)}^{m_1-1}\nabla u-\chi u{(u+1)}^{m_2-1}\nabla v\right)\\ \left(+\xi u{(u+1)}^{m_3-1}\nabla w\right)+\lambda u-\mu u^r& \text{in}\Omega \times (0,T_{max}),\\ \tau v_t=\Delta v-\varphi (t,v)+f\left(u\right)& \text{in}\Omega \times (0,T_{max}),\\ \tau w_t=\Delta w-\psi (t,w)+g\left(u\right)& \text{in}\Omega \times (0,T_{max}).\end{array}\right)$Herein, $\Omega$ is a bounded and smooth domain of $R^n$, for $n\in N$, $\chi ,\xi ,\lambda ,\mu ,r$ proper positive numbers, $m_1,m_2,m_3\in R$, and $f\left(u\right)$ and $g\left(u\right)$ regular functions that generalize the prototypes $f\left(u\right)\simeq u^k$ and $g\left(u\right)\simeq u^l$, for some $k,l>0$ and all $u\ge 0$. Moreover, $\tau \in \{0,1\}$, and $T_{max}\in (0,\infty ]$ is the maximal interval of existence of solutions to the model. Once suitable initial data $u_0\left(x\right),\tau v_0\left(x\right),\tau w_0\left(x\right)$ are fixed, we are interested in deriving sufficient conditions implying globality (i.e., $T_{max}=\infty$) and boundedness (i.e., $\Vert u(\cdot ,t){\Vert }_{L^{\infty }\left(\Omega \right)}+\Vert v(\cdot ,t){\Vert }_{L^{\infty }\left(\Omega \right)}+\Vert w(\cdot ,t){\Vert }_{L^{\infty }\left(\Omega \right)}\le C$ for all $t\in (0,\infty )$) of solutions to problem (1). This is achieved in the following scenarios:

$⊳$ For $\varphi (t,v)$ proportional to $v$ and $\psi (t,w)$ to $w$, whenever $\tau =0$ and provided one of the following conditions

(I) $m_2+k<m_3+l$, $$ (II) $m_2+k<r$, $$ (III) $m_2+k<m_1+\frac{2}{n}$

is accomplished or $\tau =1$ in conjunction with one of these restrictions

(i) $max[m_2+k,m_3+l]<r$, $$ (ii) $max[m_2+k,m_3+l]<m_1+\frac{2}{n}$,

(iii) $m_2+k<r$ and $m_3+l<m_1+\frac{2}{n}$, $$ (iv) $m_2+k<m_1+\frac{2}{n}$ and $m_3+l<r$;

$⊳$ For $\varphi (t,v)=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }f\left(u\right)$ and $\psi (t,w)=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }g\left(u\right)$, whenever $\tau =0$ if moreover one among (I), (II), (III) is fulfilled.

Our research partially improves and extends some results derived in Jiao et al. (2024); Ren and Liu (2020); Chiyo and Yokota (2022); Columbu et al. (2023).