Global properties and large-time behavior of solutions nonlinear parabolic equations

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A part of the project is devoted to the study of various classes of parabolic partial differential equations with symmetries. The basic question to be addressed is how positive solutions reflect the symmetry of the equation. For elliptic equations, there are classical theorems on symmetry of positive solutions and some motivations for symmetry problems in parabolic equations stem from these theorems (when viewing solutions of an elliptic equation as steady states of the corresponding parabolic equation). Other very interesting and challenging symmetry problems are specific to parabolic equations. Such are, for example, problems concerning the asymptotic symmetry of positive solutions as time approaches infinity. The principal investigator will continue his research in this area with the goal of obtaining a better understanding of the manner in which positive solutions approach the space of symmetric functions. Results in this vein would be instrumental in using the asymptotic symmetry of solutions to study their temporal behavior. The principal investigator will also continue his research concerning parabolic Liouville theorems. Such theorems state that certain very specific parabolic equations do not have nontrivial solutions in a class of admissible functions. Liouville theorems, when available, are very powerful tools for the qualitative analysis of parabolic equations. In combination with scaling arguments, they can be used, among other things, for the derivation of a priori estimates and for establishing blow-up and decay rates of solutions. The goal of this project is to prove new Liouville theorems and pursue further applications of Liouville theorems in various classes of nonlinear parabolic problems. Liouville theorems, as well as results on symmetry of solutions, will play important roles in another part of the project, which concerns threshold solutions. Such solutions appear as separatrices between solutions exhibiting two different kinds of behavior, such as the decay to zero and blow up in finite time, or decay to zero and locally uniform convergence to a positive steady state. Solutions of this type are studied, for example, in connection with quenching and propagation phenomena in models from combustion theory and population genetics. Up to now, existing theorems mostly treated one-dimensional equations or problems with a variational structure, thus excluding important equations that involve advection terms or explicit time dependence. The project will focus on these nonvariational problems. In less technical terms, the project can be characterized as qualitative or geometric analysis of solutions of a certain type of nonlinear evolution equations. Such equations are widely used in models in 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 improvement of their modeling relevance. 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, blow up in finite time). Development of new mathematical techniques for addressing such questions is an integral part of the project.