We consider two simple conservative systems of parabolicelliptic 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 p 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 parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L 1 ) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.Mathematics Subject Classification (2000). 35B60; 35Q80; 92C17; 92C50.

Abstract. This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel system in the full space. We derive a critical mass threshold below which global existence is ensured. Carefully using energy methods and ad hoc functional inequalities, we improve and extend previous results in this direction. The given threshold is thought to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that underlie the following approach.

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In two space dimensions, the parabolic-parabolic Keller-Segel system shares many properties with the parabolic-elliptic Keller-Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8 π. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8 π always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller-Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8 π, which is forbidden in the parabolic-elliptic case.