Blow-up dynamics for the aggregation equation with degenerate diffusion

We study radially symmetric finite-time blow-up dynamics for the aggregation equation with degenerate diffusion u(t) = Delta u(m) - del.(u*del(K*u)) in R-d, where the kernel K(x) is of power-law form vertical bar x vertical bar(-gamma). Depending on m, d, gamma, and the initial data, the solution exhibits three kinds of blow-up behavior: self-similar with no mass concentrated at the core, imploding shock solution, and near-self-similar blow-up with a fixed amount of mass concentrated at the core. Computations are performed for different values of m, d, and gamma using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement. (c) 2013 Elsevier B.V. All rights reserved.