Qualitative Properties of Solutions of Nonlinear Elliptic and Parabolic Equations

This project is concerned with nonlinear parabolic and elliptic partial differential equations (PDEs), with special focus on problems posed on an entire Euclidean space. Parabolic equations are evolution equations?the unknown function, or 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. Given an initial state of the system, the main goal is to describe its future states. Mathematically, this translates to questions about a possible development of singularities in the solutions, and, in the absence of such singularities, about the behavior of the solutions as time increases to infinity. One asks if the solution approaches in some way a time-independent steady state or if it may exhibit a more complicated behavior. Elliptic equations on Euclidean spaces represent steady states (equilibria), solitary waves, traveling fronts, or self-similar solutions of many different types of evolution PDEs. Naturally, therefore, analysis of elliptic equations is one of the key basic steps toward understanding of the dynamics of these evolution equations. The project?s problems in elliptic equations concern qualitative properties, such as symmetry, periodicity, and more complex oscillatory behavior of individual solutions, as well as the global structure of the solutions, such as their multiplicity and bifurcations (changes as parameters in the equation vary). Qualitative analysis of solutions to be carried out in this project is important for the internal development of the mathematical theory of PDEs as well as for improvement of their modeling relevance. Although the project is mainly theorical, its results concerning fundamental properties of solutions could be of interest in research fields beyond PDEs. For example, even with high computing power currently available for numerical analysis, computations involving nonlinear PDEs are often formidable without a guideline from qualitative analysis. Also, when a specific PDE model from applied science is to be investigated, general qualitative results on possible behavior of solutions of equations of the given type provide a valuable information. This project contains component projects and activities for graduate students, and the award provides graduate student research assistantship and summer support, and support for student participation at conferences.

The research in this project will develop along several main topics. In certain elliptic equations on the entire space, one of the problems concerns solutions which decay to zero in all but one variable. Employing techniques from the center manifold and KAM theories, the PI wants to examine the existence of solutions which are quasiperiodic in the non-decay variable. Another class of elliptic equations to be considered arises as an equation for self-similar solutions of the semilinear heat equation. The PI will study the multiplicity of the solutions and their bifurcations from a singular solution as the exponent in the power nonlinearity varies. For parabolic equations on the real line, the PI will continue his research on quasiconvergence properties of solutions with respect to a localized topology. For multidimensional semilinear parabolic equations on the entire space, one of the basic questions to be addressed is whether in high enough spatial dimensions, the solutions may exhibit some sort of oscillatory behavior while staying away from steady states on any sufficiently large bounded region. Two other problems deal with Liouville-type and classification theorems for entire solutions (that is, solutions defined for all times, positive and negative) of nonlinear parabolic equations. In one of them, the goal is to prove the nonexistence of positive entire solutions for an optimal Sobolev-subcritical range of exponents. In another one, the existence of nonstationary entire solutions is to be investigated for a range of supercritical exponents. Liouville and classification theorems have many interesting applications in the theory of blowup of solutions of parabolic equations and some of them, such as the characterization the type of blowup and existence of spatial blowup profiles, also belong to the expected outcome of this project.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.