Finite-time blow-up of solutions of an aggregation equation

Thomas Laurent UCLA

                     laurent@math.ucla.edu

Abstract: The aggregation equation u_{t} + div(ugradG ∗ u) = 0 arises in a number of contexts of recent interest in the physics and biology literature. Depending on how regular the potential G is, we prove either local or global existence for the solutions. For example, we will see that for G(x) = exp(−|x|) we have local existence whereas for G(x) = exp(−|x|^{2}) we have global existence. For potential with a Lipschitz point at the origin, e.g. G(x) = exp(−|x|), we prove finite-time blow-up of solutions from specific smooth initial data.