  This paper is concerned with a chemotaxis aggregation model for cells, more
precisely with a parabolic-elliptic semilinear Patlak-Keller-Segel system in a
ball of $\mathbb{R}^N$ for $N\geq 2$. For $N=2$, this system is well known for
its critical mass $8\pi$. It has been proved in \cite{Montaru2} that it also
exhibits a critical mass phenomenon for $N\geq 3$. The main result of this
paper is the exponential speed of uniform convergence of radial solutions
toward the unique steady state in the subcritical case for $N\geq 2$. We stress
that this covers in particular the classical Keller-Segel system with $N=2$,
and that the result improves on the known results even for this most studied
problem. A key tool is an associated one-dimensional degenerate parabolic
problem $(PDE_m)$ where $m$ is proportional to the total mass of cells. The
proof exploits its formal gradient flow structure $u_t=-\nabla
\mathcal{F}[u(t)]$ on an "infinite dimensional Riemannian manifold". In
particular, we show a new Hardy type inequality, equivalent to the strict
convexity of $\mathcal{F}$ at any steady state of subcritical mass, which
heuristically explains the exponential speed of convergence.
