Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model

In this paper we study the decay to the equilibrium state for the solution of a generalized version of the Goldstein-Taylor system, posed in the one-dimensional torus ${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$ , by allowing that the nonnegative cross section σ can vanish in a subregion $X:=\{ x \in {\mathbb{T}}\, \vert\, \sigma(x)=0\}$ of the domain with meas?(X)≥0 with respect to the Lebesgue measure. We prove that the solution converges in time, with respect to the strong L ²-topology, to its unique equilibrium with an exponential rate whenever $\text{meas}\,({\mathbb{T}}\setminus X)\geq0$ and we give an optimal estimate of the spectral gap.