Qualitative Properties of Solutions of Differential Equations Modeling Biological Pattern Formation

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Collapse of patterns is a newly found phenomenon characteristic to some reaction-diffusion systems possessing singular nonlinearities, where patterns are formed at first but eventually converge to a nonregular steady state. We have given sufficient conditions for patterns to collapse and also for solutions to blow-up in finite time. In addition, qualitative properties of solutions such as the dynamics of maximum points and/or asymptotic forms of solutions have been studied in detail.Moreover, movement of planar closed curves driven by bending energy is considered as a lower dimensional analogue for the geometric variational problem which determines the shape of red blood cells. All the critical points of the energy functional under some constraints are found and the gradient flow of the constraint minimization problem has been constructed.