Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

Abstract.

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 $L^{\frac{d} {2}}$ 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^p spaces with max $\left\{ {\left. {1;\frac{d} {2} - 1} \right\} \leq p < \infty .} \right.$ 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 parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L¹) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.