Abstract : 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 details 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 include semilinear reaction-diffusion systems, harmonic map heat flows, and the classical Keller-Segel system of modeling chemotaxis. I will present constructive approaches based on spectral analysis and energy-type estimates for the question of existence of blowup solutions to these particular models.