A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

Abstract

We introduce a novel gradient-based damping term into a Keller–Segel type taxis model with motivation from ecology and consider the following system equipped with homogeneous Neumann-boundary conditions:$ \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v)+a u^\alpha-b u^\beta-c|\nabla u|^\gamma & \text{ in } \Omega \times (0,T_{max}),\\ \tau v_t = \Delta v-v+u & \text{ in } \Omega \times (0,T_{max}). \end{cases} \quad\quad (\diamond) $The problem is formulated in a bounded and smooth domain $ \Omega $ of $ {\mathbb R}^N $, with $ N\geq 2 $, for some positive numbers $ a,b,c,\chi>0 $, $ \tau \in \{0,1\} $, $ \gamma\geq 1 $, $ \beta>\alpha\geq 1 $, and with $ {T_{max}}\in (0,\infty] $. As far as we know, Keller–Segel models with gradient-dependent sources are new in the literature and, accordingly, beyond giving a reasonable ecological interpretation, the objective of the paper is twofold:ⅰ) to provide a rigorous analysis concerning the local existence and extensibility criterion for a class of models generalizing problem ( $\Diamond$ ), obtained by replacing $ a u^\alpha-b u^\beta-c|\nabla u|^\gamma $ with $ f(u)-g(\nabla u) $;ⅱ) to establish sufficient conditions on the data of problem ( $\Diamond$ ) itself, such that it admits a unique classical solution $ (u,v) $ for $ {T_{max}} = \infty $ and with both $ u $ and $ v $ bounded.We handle ⅰ) whenever appropriately regular initial distributions $ u(x,0) = u_0(x)\geq 0 $, $ \tau v(x,0) = \tau v_0(x)\geq 0 $ are considered, $ f $ and $ g $ obey some regularity properties, and, moreover, some growth restrictions. Further, as to ⅱ), for the same initial data considered in the previous case, global boundedness of solutions is proven for any $ \tau\in \{0,1\} $, provided that $ \frac{2N}{N+1}<\gamma\leq 2 $.

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