LARGE-TIME BEHAVIOR OF A PARABOLIC-PARABOLIC CHEMOTAXIS MODEL WITH LOGARITHMIC SENSITIVITY IN ONE DIMENSION

This paper deals with the chemotaxis system {u(t) = Du(xx) - chi[u(ln v)(x)](x), x is an element of (0, 1), t > 0, v(t) = epsilon v(xx) + uv - mu v, x is an element of (0, 1), t > 0, under Neumann boundary condition, where chi < 0, D > 0, epsilon > 0 and mu > 0 are constants. It is shown that for any sufficiently smooth initial data (u(0), v(0)) fulfilling u(0) >= 0, u(0) not equivalent to 0 and v(0) > 0, the system possesses a unique global smooth solution that enjoys exponential convergence properties in L-infinity(Omega) as time goes to infinity, which depend on the sign of mu - (u) over bar (0), where (u) over bar (0) := integral(1)(0) u0dx. Moreover, we prove that the constant pair (mu, (mu/lambda)(D/chi)) (where lambda > 0 is an arbitrary constant) is the only positive stationary solution. The biological implications of our results will be given in the paper.