Abstract
In this paper, we consider the following dual-gradient chemotaxis model The model was proposed to interpret the spontaneous aggregation of microglia in Alzheimer's disease due to the interaction of attractive and repulsive chemicals released by the microglia. It has been shown in the literature that, when m = 1, the solution of the model with homogeneous Neumann boundary conditions either blows up or asymptotically decays to a constant in multi-dimensions depending on the sign of = χα - ξγ, which means there is no pattern formation. In this paper, we shall show as m > 1, the uniformly-in-time bounded global classical solutions exist in multi-dimensions and hence pattern formation can develop. This is significantly different from the results for the case m = 1. We perform the numerical simulations to illustrate the various patterns generated by the model, verify our analytical results and predict some unsolved questions. Biological applications of our results are discussed and open problems are presented.
| Original language | English |
|---|---|
| Pages (from-to) | 307-338 |
| Number of pages | 32 |
| Journal | Analysis and Applications |
| Volume | 16 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 1 May 2018 |
Keywords
- attraction-repulsion
- boundedness
- chemotaxis
- Dual-gradient
- higher dimensions
- pattern formation
- spontaneous aggregation
ASJC Scopus subject areas
- Analysis
- Applied Mathematics