| Abstract: |
| The Gierer-Meinhardt system is a model for pattern formation based on Turing`s mechanism. Under some conditions, it reduces to a scalar heat equation, with a nonlinearity showing a pure power divided by some non-local term. Depending on parameters, that equation shows two different types of blow-up behavior:
- in some subcritical range of parameters, the non-local term converges to a positive constant, leading to some blow-up behavior similar to the classical semilinear heat equation, with power nonlinearity ;
- in the critical case, the non-local term converges to infinity, weakening the effect of the pure power nonlinearity. This leads to a new type of blow-up behavior, unknown in earlier literature.
In this talk, we construct examples for the two types of behaviors, and give their blow-up profiles. Our method happens to be a non-trivial adaptation of the classical construction method for the semilinear heat equation. |
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