We are concerned with deriving sharp exponential decayestimates (i.e. with maximum rate and minimum multiplicative constant )for linear, hypocoercive evolution equations. Using a modal decomposition ofthe model allows to assemble a Lyapunov functional using Lyapunov matrixinequalities for each Fourier mode.We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest Fouriermodes determine the spectral gap of the whole equation inL2. By contrast,on the whole real line the Goldstein-Taylor model does not have a spectralgap, since the decay rate of the Fourier modes approaches zero in the smallmode limit. Hence, the decay is reduced to algebraic. In the final part of the talk we consider the Goldstein-Taylor model withnon-constant relaxation rate, which is hence not amenable to a modal decom-position. In this case we construct a Lyapunov functional of pseudodifferen-tial nature, one that is motivated by the modal analysis in the constant case.The robustness of this approach is illustrated on a multi-velocity Goldstein-Taylor model, yielding explicit rates of convergence to the equilibrium.This is joint work with J. Dolbeault, A. Einav, C. Schmeiser, B. Signorello, and T. Wöhrer. -----------------------------------------------------------------------------References [1] A. Arnold, A. Einav, B. Signorello, T. W ̈ohrer: Large-time convergenceof the non-homogeneous Goldstein-Taylor equation, J. Stat. Phys. 182(2021) 41.[2] A. Arnold, J. Dolbeault, C. Schmeiser, T. W ̈ohrer: Sharpening of decayrates in Fourier based hypocoercivity methods, To appear in INdAMproceedings (2021).