Qualitative Studies of Nonlinear Elliptic and Parabolic Equations

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This project is concerned with nonlinear parabolic and elliptic partial differential equations. Parabolic equations are evolution equations--the unknown function (i.e., the solution) depends on one or several spatial variables and one more distinguished variable playing the role of time. Such equations are widely used in models in applied sciences, in particular, in chemical engineering, combustion theory, and ecology. Given an initial state of the system, the problem is to describe its future states. Mathematically, this translates to an understanding of the spatial structure (e.g., homogeneity, symmetry, concentration) of the solution at large times, as well as of its temporal behavior, such as approach to a time-independent steady state or periodic behavior, or possibilities of an even more complicated behavior. Elliptic equations are equations whose solutions can be viewed as time-independent solutions, or equilibria, of parabolic equations (and many other types of evolution equations). Naturally, therefore, analysis of elliptic equations is one of the key basic steps toward understanding the dynamics of parabolic equations. Of particular significance to the present project are symmetry properties of steady states and the global structure of the whole set of steady states for certain elliptic equations. Qualitative analysis of solutions to be carried out in this project is important for the internal development of the mathematical theory of partial differential equations as well as for improvement of their modeling relevance. Rigorous analysis maintains its indispensable role even in the presence of the high computing power currently available for numerical analysis. Not only does it provide guidelines for and simplifications of otherwise formidable computations, in many situations qualitative analysis is the only way to deal with difficult problems concerning general solutions of nonlinear equations. The research in this project will develop along several main topics. For parabolic equations on the real line, the principal investigator will first analyze the behavior of front-like solutions and their approach to propagating terraces (stacked systems of traveling fronts). He will then take a closer look at quasiconvergence properties of general solutions with respect to a localized topology. For multidimensional parabolic problems on the entire space, one of the basic questions to be addressed is whether bounded solutions converge to equilibrium, at least along a sequence of times, as solutions of the one- and two-dimensional equations do. Two other problems deal with Liouville-type theorems for entire solutions of nonlinear parabolic equations. In one of them, the principal investigator suggests a way of using a Liouville theorem in a proof of the approach to propagating terraces for solutions of multidimensional parabolic problems. In the other one, scaling techniques in parabolic partial differential equations and a Liouville theorem are used for analyzing solutions with singularities. A major problem in this area is to determine the optimal range of exponents for the validity of the Liouville theorem. In elliptic equations on the entire space, one of the problems concerns solutions that decay to zero in all but one variable. The principal investigator seeks to establish the existence of solutions that are quasiperiodic in the nondecay variable. He will also continue working on his projects on symmetry and the nodal structure of nonnegative solutions of elliptic and parabolic equations and on threshold solutions in various parabolic problems.