Sharp traveling waves for degenerate diffusion equations with delay

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This research proposal concerns the study of partial differential equations arising from ecology, a class of degenerate diffusion equations with delay, which describe the population dynamics of the single species with age-structure, such as Australian blowflies. Degenerate diffusion caused by competition between conspecifics or deteriorating environmental conditions is a common feature of population spreading modeling in ecology, however, there are numerous gaps in our knowledge about the wave dynamical behaviors of degenerate diffusion equations compared with the well-studied linear diffusion case. The degeneracy usually causes the solutions to lose their regularities, and raises the possibility of sharp type traveling waves, where the population density decreases to zero at a finite point, rather than decaying to zero asymptotically. On the other hand, the large time-delay often causes the solutions to be oscillating. These two facts make the structure of solutions more complicated and the relevant study more challenging.    The purpose of this research project is to understand the effects of time-delay and degeneracy of diffusion for the population dynamic models. The main objectives include: One is about the structure of traveling waves. Affected by the time-delay and degeneracy of diffusion, we are particularly interested in the classification of sharp traveling waves, oscillatory traveling waves, and oscillatory sharp wavefronts. We will try to prove the existence/non-existence of these waves, and clarify what will be the criteria and mechanism for the population dynamic system to possess these wavefronts. The second issue is to try to study the stability of these wavefronts, in particular, the sharp waves and oscillating sharp waves. Because, the structure of sharp waves is totally different from the monotone/non-monotone traveling waves, and the asymptotic stability of these wavefronts is quite challenging and never treated.    The proposed research program is expected to advance and enrich the theory of degenerate diffusion equations with time-delay, to introduce new mathematical ideas and methods to study stability of sharp wavefronts, and also to provide good opportunity to train Ph.D. students and postdocs by carrying out this project.