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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.

35Q20 Boltzmann equations
35F10 Initial value problems for linear first-order PDEs
Full Text: DOI
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