Qualitative studies of solutions of nonlinear elliptic and parabolic equations

The project is devoted to qualitative studies of partial differential equations (PDE). Building on recent developments, the principal investigator will study symmetry properties and the nodal structure of nonnegative solutions of elliptic PDE. The main goals are to classify spatial domains on which solutions with nontrivial nodal sets can exist and to determine whether such solutions can exist at all for spatially homogeneous equations. In parabolic equations, a tendency of positive solutions to 'improve their symmetry' as time increases to infinity is a remarkable example of how parabolic flows can reduce spatial complexity. The principal investigator will continue his study of this interesting asymptotic symmetry phenomenon, while bearing in mind applications of asymptotic symmetry theorems in convergence results for parabolic equations. Other methods will also be employed to address several long-standing problems concerning the convergence of solutions of parabolic PDE to equilibria. Another topic in this project concerns positive solutions of elliptic PDE on the whole Euclidean space that decay to zero in some variables but do not decay in other variables. The symmetry of the solutions with respect to the decay variables and their behavior with respect to the remaining variables will be examined. The principal investigator will also continue his research concerning Liouville-type theorems on the nonexistence of nontrivial solutions for specific classes of nonlinear equations. Scaling techniques based on Liouville theorems have a wide range of applications in the theory of parabolic PDE, which will be further explored in the project. Results of the above projects will be applied in studies of threshold solutions in various parabolic problems. Such solutions occur as separatrices between solutions exhibiting two different kinds of behavior, such as the decay to zero and blow-up in finite time. They have been studied for purely theoretical reasons as well as in connection with quenching and propagation phenomena in applied sciences.

In less technical terms, the project can be characterized as qualitative or geometric analysis of solutions of nonlinear partial differential equations. Such equations are widely used in models in the applied sciences, in particular, chemical engineering, combustion theory, and ecology. Understanding qualitative properties of solutions is important for the internal development of the mathematical theory of partial differential equations as well as for the improvement of their modeling relevance. For the interpretation of models involving nonlinear partial differential equations, rigorous analysis maintains its indispensable role even in presence of the high computing power currently available for numerical analysis. Not only does it provide guidelines for and simplifications of otherwise formidable computations, but in many situations qualitative analysis is the only way to deal with difficult problems concerning general solutions of nonlinear equations. The present project addresses questions that concern geometric properties of solutions (such as their symmetries when viewed as functions of spatial variables) as well as their behavior with respect to time (periodicity properties, stabilization to equilibria, so-called blow-up in finite time). Development of new mathematical techniques for addressing such questions is an integral part of the project.