Nonlinear evolution equations, i.e., nonlinear time-dependent partial differential equations are central to
mathematical description of natural phenomena. For example, 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. What is more,
these equations play an important role in the treatment of pure mathematical problems as well.
In this project we are concerned with two types of evolution equations, namely the wave and the
parabolic ones. They 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. However, in spite of their ubiquity in applications, the mathematical understanding of these
equations is still unsatisfactory. A question that is particularly relevant yet underexplored 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., formation of a black hole), or the
emergence of a new (singular) structure (e.g., drop formation in liquids), or that in fact some essential
physics is missing from the model.
Recently, there has been a significant research progress on the topic of blowup for wave and parabolic
equations. However, most of the results concern the so-called critical and subcritical cases, i.e., the
instances where conserved quantities of the equations can be used to control the evolution. If such control
is not possible then the equations are referred to as supercritical, and their analysis is traditionally more
difficult. What is more, numerical studies of various types of supercritical equations indicate that large
data solutions generically break down, with blowup being governed by the so-called self-similar solutions.
In view of these insights, the main aim of this project is the development of 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, the hyperbolic Yang-Mills equations 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.