Global existence and uniqueness of solutions in a chemotaxis system with density-suppressed motility, a $θ-$logistic source, and indirect signal production under a strong Allee effect

Abstract

We consider the following chemotaxis-growth system, featuring density-suppressed motility and a $θ$-logistic growth term that incorporates a strong Allee effect: \begin{equation*} \left\{ \begin{aligned} u_t &= Δ(γ(v)u) + μu(1-u^θ)(u-A), & x \in Ω, \quad t > 0, v_t &= Δv - v+ w^β, & x \in Ω,\quad t > 0, w_t &= -δw +u, & x \in Ω, \quad t > 0, \end{aligned} \right. \end{equation*} subject to homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 2)$ with smooth boundary. Here $μ\in \mathbb{R} , δ,β>0, θ\geq1$, and the positive motility function $γ(v) \in C^3([0,\infty))$ fulfills $γ'(v) \leq 0$ for all $v\geq 0$. The main objective of this paper is to establish the global existence and boundedness of classical solutions to the proposed problem. Our analysis combines the Schauder fixed point theorem for proving local-in-time existence with the extensibility criterion, fundamental energy estimates, $L^p-$bounds, the Gagliardo-Nirenberg interpolation inequality, Young's inequality, semigroup estimates, and a Moser-type iteration scheme to derive uniform-in-time $L^{\infty}-$bounds, thereby ensuring global bounded classical solutions. More precisely, we establish that the chemotaxis-growth system admits a unique globally bounded classical solution whenever $β< \frac{2(θ+2)}{d}$. Furthermore, if the logarithmic derivative of the motility function is assumed to be uniformly bounded on $[0,\infty)$, that is, $\frac{γ'(v)}{γ(v)} \in L^{\infty}([0,\infty))$, then the above restriction on $β$ can be weakened to $β< \frac{d(θ+1)+2(θ+2)}{2d}$, provided that $θ> \frac{4-d}{d-2}$.