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Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model. (English) Zbl 1291.35157
Summary: 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 \(\sigma\) can vanish in a subregion \(X:=\{x\in\mathbb T\mid\sigma (x)=0\}\) of the domain with meas \((X)\geq 0\) 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 meas \((\mathbb T\setminus X)\geq 0\) and we give an optimal estimate of the spectral gap.

MSC:
35Q20 Boltzmann equations
35F10 Initial value problems for linear first-order PDEs
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