  We study the global existence of solutions to a one-dimensional
drift-diffusion equation with logistic term, generalizing the classical
parabolic-elliptic Keller-Segel aggregation equation arising in mathematical
biology. In particular, we prove that there exists a global weak solution, if
the order of the fractional diffusion $\alpha \in (1-c_1, 2]$, where $c_1>0$ is
an explicit constant depending on the physical parameters present in the
problem (chemosensitivity and strength of logistic damping). Furthermore, in
the range $1-c_2<\alpha\leq 2$ with $0<c_2<c_1$, the solution is globally
smooth. Let us emphasize that when $\alpha<1$, the diffusion is in the
supercritical regime.
