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 = Δ u m − ∇ ⋅ ( u ∗ ∇ ( K ∗ u ) ) in R d , where the kernel K ( x ) is of power-law form | x | − γ . Depending on m , d , γ , 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 γ using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement.