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Research Article

Convergence towards equilibrium for a model with partial diffusion

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Received 19 Jan 2023, Accepted 13 Apr 2024, Published online: 09 May 2024
 

Abstract

We study the asymptotic behavior of a two dimensional linear PDE with a degenerate diffusion and a drift term. The structure of this equation typically arises in some mathematical mean-field models of neural network, and the investigation of the qualitative properties of this equation is still open, and a challenging question. We prove, via a Doeblin-Harris type method, that the solutions converge exponentially fast to the unique stationary state in a L1-weighted norm.

Notes

1 Strictly speaking, the computations in Lemma 4.1 should have first been restricted to smooth initial data u0, and then extended by continuity as here, we have been voluntarily a bit sloppy for the ease of reading.

Additional information

Funding

Delphine Salort is supported by the ANR project ChaMaNe (ANR-19-CE40-0024). Didier Smets is supported by the ANR project ODA (ANR-18-CE40-0020-01).

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