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.