Mathematical Sciences: Nonlinear Partial Differential Equations and Systems

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Ni 9401333 Work supported by this award focuses on the mathematical analysis of nonlinear partial differential equations, systems and their applications. The research continues several projects such as the study of spike-layers in mathematical biology. These arise in mathematical descriptions of the chemotactic aggregation stage of cellular slime molds. One is interested in solutions of the relevant equations which exhibit spiky patterns dependent on the equation coefficients. Such patterns have been observed experimentally. A second line of investigation concerns cross- diffusion in population dynamics. The primary question to be studied here is one of determining and understanding the steady states of such systems. Work will also be done measuring the stability of steady states of a class of semilinear heat equations in n-dimensional space and on the inhomogeneous semilinear elliptic or parabolic equations arising in infinite particle systems in probability. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.