| Abstract: |
| As a model of pattern formation in hydra, Gierer and Meinhardt proposed an activator-inhibitor system. Numerical simulations show that this system produces spiky patterns such that the distribution of solutions concentrates in a very narrow region around finitely many points. In the case of spatially uniform equations rigorous results have been obtained on the existence and stability of spiky patterns. In this talk we are interested in such concentration phenomena for a spatially heterogeneous reaction-diffusion equation. In particular, to study the effects of spatial heterogeneity on concentration points, we introduce a locator function composed only of the coefficients in the equation and prove that any concentration point must be a critical point of the locator function. Moreover, we construct a solution concentrating near a nondegenerate critical point of the locator function. |
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