Analysis of linear and nonlinear wave phenomena and the inverse problem

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In this project, we are mainly concerned with the analysis of wave phenomena governed by some basic partial differential equations in Applied Mathematics and Physics, and also by some model equations appearing in Chemistry and Biology. Summarizing the results obtained by the investigators in the period 200 1-2003, we can say the objective of this project is accomplished fruitfully.The head investigator published 5 papers analyzing linear and nonlinear waves and nonlinear diffusions. The topics include the following:(1)Nonlinear parabolic equations: Large time asymptotics of solutions for generalized KPP equation are studied. Precise formula obtained here is expected to have many applications in Biology.(2)Inverse spectral problem: Inverse problem to determine the potential from some spectral data is studied. We obtained a uniqueness results for Sturm-Liouville operator and for Dirac operator on firiete interval. Our problem 'to determine the potential from interior spectral data' is a new formulation and is expected to be applied to many other operators.(3)Semilinear wave equations: Asymptotic for wave equation with nonlinear dissipation is studied. We require that the dissipation is inhomogeneous in' space and time and sufficient conditions are obtained for solutions to be asymptoically free.(4)Inverse scattering problem: First the direct scattering theory is established for wave equation with first time derivative term, and then the coefficient of the term is shown to be reconstructed from the scattering amplitude with a fixed energy.Each investigator developed many interesting results on the related fields.