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We consider the system of equations given by
\begin{eqnarray}\label{eqa1}
\partial_t \rho & = & \beta(\Delta)^{\alpha/2}\rho+\nabla\cdot(\nabla u\rho)\\ \label{eqa2}
\Delta u & = & \rho\langle \rho \rangle.
\end{eqnarray}
This system of pde was proposed by Keller and Segel as a model for the motion of cells. It is assumed that these cells ($\rho$) move towards increasing concentrations of some chemical substance ($u$). The system \eqref{eqa1}\eqref{eqa2} also appears as a model of gravitational collapse. Indeed the system \eqref{eqa1}\eqref{eqa2} is very similar in spirit to the Zel'dovich approximation used in Cosmology to study the formation of largescale structure in the primordial universe.
Motivated by these problems, we addressed the analysis of \eqref{eqa1}\eqref{eqa2}. In particular we study the continuation criteria and the global existence of classical solution for small initial data. These results can be found in (Ascasibar,Granero,Moreno,\emph{Physica D}, 2013). Furthermore, we study the onedimensional, quasilinear, critical system
\begin{eqnarray}\label{eqa1b}
\partial_t \rho & = & \partial_x(\beta(\rho)H\rho)+\partial_x(\partial_x u\rho)\\ \label{eqa2b}
\partial_x^2 u& = & \rho\langle \rho \rangle,
\end{eqnarray}
where $H$ denotes the Hilbert transform. We obtained global existence of weak solution, among other results. These results can be found in (Granero,Orive,\emph{Submitted},2013). During this talk, we will try to motivate and explain the main results in these two papers. 
