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Monday, December 16
Tuoc Van Phan, The University of Tennesse, Knoxville
Some Aspects in Nonlinear Partial Differential Equations and Nonlinear Dynamics
Computer Science and Mathematics Division Seminar
2:00 PM — 3:00 PM, Research Office Building (5700), Room L-204
Contact: Cory Hauck (firstname.lastname@example.org), 865.574.0730
AbstractPart I: We discuss the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. We show that if there are self-diffusion in one species and no cross-diffusion in the other, then the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global W ^(1,p) –estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.
Part II: We study a class of nonlinear Schrödinger equations in one dimensional spatial space with double-well symmetric potential. We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time-dependent solutions of the nonlinear Schrödinger equation over long but finite time intervals.