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 T = R/Z, by allowing that the nonnegative cross section σ can vanish in a subregion X:= {x ∈ T{pipe} σ(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 L²-topology, to its unique equilibrium with an exponential rate whenever (T\X)≥0 and we give an optimal estimate of the spectral gap.