Mathematical Sciences: Topics in Partial Differential Equations and Their Applications

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9625830 Wang The proposer plans to conduct his research in the following topics: (i) Convolution models for phase transitions; (ii) Concentration phenomena of solutions of semilinear elliptic equations; (iii) Qualitative behavior of solutions to chemotaxis systems. In the first front, effort will be made to understand the propagation of interfaces, the shape of energy minimizers and the dynamics of typical solutions. In the second front, of main concern are the existence and locations of concentration of solutions as a physical parameter becomes small. In the third front, the effects of various biological and physical parameters on the solutions of the systems will be studied. %%% The convolution models arise in the mathematical studies of grain boundaries in crystalline solids, binary alloys as well as in the study of excitation in neural networks. The equations in the second front are nonlinear Schr\"odinger equations of mathematical physics and one arising in the study of chemotaxis of cells. The chemotaxis systems considered in the third front model the oriented motion of cells in response to the concentration gradient of chemicals in the environment, and the research done and proposed here aims at understanding how the motility and the sensitivity of cells to the chemical affect the size of the population of cells as well as the way in which cells are distributed in the long run. The proposed research in all the three parts will help reveal and understand the important phenomena such as phase seperations, concentration and blow-up in the related fields of material sciences, biophysics and quantum mechanics. ***