Abstract:In this talk, we mainly study non-negative classical solutions to a Neumann initial-boundary value problem for a higher dimensional parabolic elliptic ODE minimal chemotaxis-haptotaxis system. We first show pure haptotaixs cannot induce any blow-up and pattern in any dimensions, showing negligibility of haptotaxis on boundedness. Then, in the radial setting, it is well-known that small mass of cells can lead to blow-up in the corresponding Keller-Segel chemotaxis-only model, known as generic mass blow-up phenomenon. Herein, in the presence of the temporal nonlocality brought by haptotaxis, we show, in an explicit realm, that the aforementioned generic mass finite time blow-up phenomenon preserves. This seems to be the first rigorous blow-up result in relevant chemotaxis-haptotaxis models. The blow-up result also suggests that haptotatic cross-diffusion may not be negligible compared to chemotactic aggregation in three or higher D, in contrast to the recently detected 2D negligibility of haptotaxis in a minimal chemotaxis haptotaxis model.
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