Many central problems in geometry, mathematical physics and biology reduce to questions regarding the behavior of solutions of nonlinear evolution equations. The global dynamical behavior of bounded solutions for large times is of significant interest. However, in many real situations, solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities or to understand their geometry in conjunction with globally bounded solutions. In this context we have been particularly interested in qualitative descriptions of blowup. Particular examples in the talk include non-variational semilinear parabolic systems, wave maps for which the sole existence of blowup solutions in super-critical regimes has been a long standing open problems, and the classical Keller-Segel system of modeling chemotaxis. I will present different techniques based on spectral analysis or/and energy methods to study the question of existence and stability of blowup solutions to these equations.
Van Tien Nguyen received his PhD from the University Paris 13, France in December 2014 and has been working at NYU Abu Dhabi since January 2015. He does research on Partial Differential Equations (PDEs) with focusing on singularity formation and long-time dynamics of solutions. Some classes of equations he has studied are Semilinear Reaction-Diffusion systems, geometric evolution equations and Nonlinear Aggregation-Diffusion equations.