Finite-time blow-up of $ L^{\infty}$ Weak solutions of an aggregation equation

Jeremy Brandman
Department of Mathematics

Abstract: In recent years, the integrodifferential equation $ u_t + \nabla \cdot [(\nabla K)\ast u)u] = 0$ has been used to model biological aggregation and dispersion. During this talk, we discuss recent work on existence, uniqueness, and finite-time blow-up of solutions to this equation for nonnegative initial data belonging to $ L^1 \int L^{\infty}$. For kernels $ K$ which are rotationally invariant, nonnegative, and decay at infinity, finite-time blow-up is proven when $ K$ has a Lipschitz point at the origin.