Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources

Tahir Bachar Issa; Wenxian Shen

paper

Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources

arXiv:1609.00794

Abstract

This paper considers the dynamics of the following chemotaxis system $⎩ ⎨ ⎧ u_{t} = Δ u - χ \nabla (u \cdot \nabla v) + u (a_{0} (t, x) - a_{1} (t, x) u - a_{2} (t, x) \int_{Ω} u), x \in Ω 0 = Δ v + u - v, x \in Ω \frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0, x \in \partial Ω,$ where $Ω \subset R^{n} (n \geq 1)$ is a bounded domain with smooth boundary $\partial Ω$ and $a_{i} (t, x)$ ( $i = 0, 1, 2$ ) are locally Hölder continuous in $t \in R$ uniformly with respect to $x \in Ω ˉ$ and continuous in $x \in Ω ˉ$ . We first prove the local existence and uniqueness of classical solutions $(u (x, t; t_{0}, u_{0}), v (x, t; t_{0}, u_{0}))$ with $u (x, t_{0}; t_{0}, u_{0}) = u_{0} (x)$ for various initial functions $u_{0} (x)$ . Next, under some conditions on the coefficients $a_{1} (t, x)$ , $a_{2} (t, x)$ , $χ$ and $n$ , we prove the global existence and boundedness of classical solutions $(u (x, t; t_{0}, u_{0}), v (x, t; t_{0}, u_{0}))$ with given nonnegative initial function $u (x, t_{0}; t_{0}, u_{0}) = u_{0} (x)$ . Then, under the same conditions for the global existence, we show that the system has an entire positive classical solution $(u^{*} (x, t), v^{*} (x, t))$ . Moreover, if $a_{i} (t, x)$ $(i = 0, 1, 2)$ are periodic in $t$ with period $T$ or are independent of $t$ , then the system has a time periodic positive solution $(u^{*} (x, t), v^{*} (x, t))$ with periodic $T$ or a steady state positive solution $(u^{*} (x), v^{*} (x))$ . If $a_{i} (t, x)$ $(i = 0, 1, 2)$ are independent of $x$ , then the system has a spatially homogeneous entire positive solution $(u^{*} (t), v^{*} (t))$ . Finally, under some further assumptions, we prove that the system has a unique entire positive solution $(u^{*} (x, t), v^{*} (x, t))$ which is globally stable . Moreover, if $a_{i} (t, x)$ $(i = 0, 1, 2)$ are periodic or almost periodic in $t$ , then $(u^{*} (x, t), v^{*} (x, t))$ is also periodic or almost periodic in $t$ .