Finite-Time Blow-Up of Solutions of an Aggregation Equation

Thomas Laurent

Abstract: The aggregation equation $u_t + div(u grad G * u )
= 0$ 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 $G(x)=exp(-\vert x\vert)$ we have local existence whereas for $G(x)=exp(-\vert x\vert^2)$ we have global existence. For potential with a Lipschitz point at the origin, e.g. $G(x)=exp(-\vert x\vert)$, we prove finite-time blow-up of solutions from specific smooth initial data.