Dynamics of the convergence towards a self-similar blowup solution in a simplified model of aggregation

Abstract

The dynamics of the finite-time blowup solutions of a parabolic–elliptic system of partial differential equations is studied. These equations arise when modelling chemotactic aggregation or a dissipative gravitational collapse. Radial self-similar blowup solutions on a bounded domain are analysed by perturbing the known analytic solutions of the corresponding unbounded problem. The dynamics followed by general initial conditions leading to these blowup solutions is studied numerically. They are shown to converge to the self-similar profile in a non-uniform way. In similarity coordinates (where self-similar blowup solutions appear as stationary), their convergence properties are characterized by the eigensystem associated to the linearized time evolution equations. The resulting eigenvalues λn and eigenvectors are presented for various values of the space dimension parameter d. The asymptotic behaviours of λn are found for d→2 and for large d. A simple numerical formulation for this problem, obtained by reparametrizing the blowup profile dynamics, is presented in the appendix. It simplifies the numerical task by reducing the number of resolution points needed to describe the blowup profile when approaching the singularity.