Eventual smoothness and stabilization of renormalized radial solutions in a chemotaxis consumption system with bounded chemotactic sensitivity

In this paper, a chemotaxis model with bounded chemotactic sensitivity and signal absorption {u(t) = Delta u - del center dot )uS(u, v)del u), x is an element of Omega, t > 0, vt = Delta v - uv, x is an element of Omega, t > 0, (star) is considered under homogeneous Neumann boundary conditions in the ball Omega = B-R(0) subset of R-n, where R > 0 and n >= 2. Here, S is a scalar function with S(s, t) is an element of C-2([0, infinity) x [0, infinity)). Moreover, for some positive constant K, vertical bar S(s, t)vertical bar <= K for all s, t is an element of [0, infinity). For all appropriately regular and radially symmetric initial data (u(0), v(0)) fulfilling u(0) >= 0 and v(0) > 0, the present paper shows that there is a globally defined pair (u, v) of radially symmetric functions which are continuous in ((Omega) over bar\{0}) x[0, infinity) and smooth in ((Omega) over bar\{0}) x(0, infinity), and which solve the corresponding initial-boundary value problem for (star) with (u(center dot, 0), v(center dot, 0)) = (u(0), v(0)) in an appropriate generalized sense. Moreover, in the two-dimensional setting, it is shown that these solutions are global mass-preserving in the flavor of the identity integral(Omega) u(x, t) = integral(Omega) u(0)(x) for all t > 0 and any nontrivial of these globally defined solutions eventually becomes smooth and satisfies u(center dot, t) -> 1/vertical bar Omega vertical bar integral(Omega) u(0), and v(center dot, t) -> 0 as t -> infinity uniformly with respect to x is an element of Omega.