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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 large-scale 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 one-dimensional, 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. |
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