The project will develop along the following central themes: symmetry properties of positive solutions of parabolic PDEs, principal Floquet bundles and exponential separation, large time behavior of solutions of parabolic PDEs and the dynamics of quasilinear competition-diffusion systems.
Symmetry (radial or reflectional) plays an important role in qualitative analysis of solutions of PDEs. By now, a fairly general understanding of symmetry of positive solutions has been achieved for second order elliptic equations on both bounded and unbounded domains, and for parabolic equations on bounded domains. For nonautonomous parabolic equations on unbounded domains the symmetry problem has not been addressed so far. This problem is more intriguing; standard methods used in earlier symmetry results do not give desired conclusions. Exponential separation, associated with the principal Floquet bundle of linearized parabolic equations, is among new techniques that can facilitate the analysis. The principal Floquet bundle is an interesting topic in its own right. It is a natural extension of the concept of principal eigenfunction of elliptic operators to nonautonomous parabolic operators which has already proved very useful in the study of nonlinear equations. Both its basic properties and applications are to be further investigated. A better understanding is needed to make this tool applicable to problems on unbounded domains, however, ideas related to this concept have already triggered progress in the symmetry problem. The next topic, large-time behavior, includes in particular a basic question on the semilinear heat equation, as to whether all bounded solutions on multidimensional domains converge to an equilibrium. This has been already been answered (negatively) for spatially inhomogeneous equations. For homogeneous equations, this long standing open problem will require new ideas. Competition-diffusion systems that are to be considered in the project have their origins in ecology. Bifurcations of steady states, their stability and as complete as possible an understanding of global dynamics is expected to shed light on coexistence of species (of phenotypes) with different dispersal rates. While a large class of semilinear systems has already been successfully treated, more realistic quasilinear systems call for a new approach.
In less technical terms, the project can be characterized as qualitative or geometric analysis of solutions of nonlinear evoution equations, such as reaction-diffusion equations. As a rule, such nonlinear equations can "never be solved". Letting aside exact solutions which are very rarely available, approximate methods, even with the presently available computation power, can seldom provide answers to questions of global nature, for example, questions on the behavior of solutions on infinite time intervals or collective behavior of solutions (structural stability). Yet, for the internal development of the theory of PDEs as well as for improvement of their modeling relevance in other sciences, such questions are important to ask and answer. For this purpose, many methods of qualitative analysis of nonlinear PDEs have been developed, mainly in the last 2-3 decades. The present project relies on these methods, combining classical PDE techniques with dynamical systems ideas, and, at the same time, attempts at development of new techniques. Among main objectives of the project is the description of the large time behavior of the solutions. Spatial profiles (symmetries) and temporal behavior (asymptotic periodicity, stabilization to equilibria) will both be examined.
