*Mathematical Methods in the Applied Sciences*Analysis of long-term solution of chemotactic model with indirect signal
consumption in three-dimensional case

In this paper, we consider the chemotaxis model
u_t&=\Delta
u-\nabla\cdot(u\nabla v),&
\qquad
x\in\Omega,\,t>0,v_t&=\Delta
v-vw,& \qquad
x\in\Omega,\,t>0,w_t&=-\delta
w+u,& \qquad
x\in\Omega,\,t>0,under
homogeneous Neumann boundary conditions in a bounded and convex domain
$\Om\subset
\mathbb{R}^3$ with smooth boundary, where
$\delta>0$ is a given parameter. It is
shown that for arbitrarily large initial data, this problem admits at
least one global weak solution for which there exists
$T>0$ such that the solution $(u,v,w)$ is bounded and
smooth in
$\Om\times(T,\infty)$.
Furthermore, it is asserted that such solutions approach spatially
constant equilibria in the large time limit.

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