Global Solutions of some Chemotaxis and Angiogenesis Systems in high space dimensions

We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the L d 2 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the L spaces with max{1; d 2 − 1} ≤ p < ∞. This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolicdegenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.