Self-similar blowup for supercritical evolution equations

Nonlinear evolution equations, i.e., nonlinear time-dependent partial differential equations, are central to mathematical description of natural phenomena. Fluid flow and the evolution of the universe in physics, population dynamics and cell division in biology, cancer growth and spreading of infectious diseases in medical sciences are all modeled by means of nonlinear evolution equations. In addition, these equations play an important role in the treatment of pure mathematical problems as well. A particularly relevant notion in this context is the one of finite time breakdown of solutions (also called singularity formation or blowup). Physically, singularity formation indicates a radical change in the modeled phenomenon (e.g., breaking of water waves or implosion of a star), or the emergence of a new (singular) structure (e.g., formation of drops and bubbles in liquids), or that in fact some essential physics is missing from the model. In line with this, it is of utmost importance to understand the existence of blowing up solutions as well as to study their stability which is a key physical requirement for their observability. Mathematically, an often-used concept to capture universal properties of blowup is the one of self- similarity. Indeed, numerical studies of various types of equations of the so-called supercritical type indicate that large data solutions generically break down in a self-similar manner. In view of these insights, in this project, which is a continuation of the project P 34378, we are concerned with the question of the existence and stability of self-similar blowup for two types of evolution equations, namely the wave and the parabolic ones. (These equations arise in a number of areas from sciences and engineering, e.g., fluid dynamics, optics, general relativity, particle physics, population dynamics, image processing, heat conduction, and many more.) More precisely, the aim of this project is to develop new, general and robust methods for the study of the existence and stability of self-similar blowup for supercritical wave and parabolic equations. We intend to do this through the analysis of concrete models: the focusing power nonlinearity wave and heat equations, wave maps, and the Keller-Segel model for chemotaxis. Our approach entails a combination of methods from the classical analysis of partial differential equations, ordinary differential equations theory, nonlinear functional analysis, operator theory and approximation theory. Our overarching aim is a systematic and rigorous study of stability of blowup in evolution equations in general, and development of techniques that can be efficiently applied to realistic physical models.

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