| Abstract: |
| Nonlinear partial differential equations (PDEs) arise naturally in many biological and chemical systems, particularly in the context of cross-diffusion systems such as chemotaxis. Additionally, random fluctuations are prevalent in the real world, and this randomness can lead to various new phenomena that significantly impact the behavior of the solutions. The introduction of a stochastic term (or noise) in the model often results in qualitatively new behaviors, enhancing our understanding of real processes and often making them more realistic.
The interplay between noise and nonlinearity can give rise to effects such as noise-induced transitions, stochastic resonance, metastability, or even noise-induced chaos.
In the case of simple Keller-Segel models, a dichotomy exists based on the initial mass; depending on this mass, a blow-up phenomenon may occur. However, it is possible to suppress this blow-up by introducing an additional term. When incorporating some multiplicative noise, the mass may change over time. In this scenario, it is also possible to demonstrate the existence of a global solution if a proliferation term or a porous media term is included. In the talk, we will introduce some Keller-Segel systems for which a global solution exists even in the two-dimensional case. |
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