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A new framework to obtain time-decay estimates for partially dissipativehyperbolic systems set on the real line is developed. Under the classicalShizuta-Kawashima (SK) stability condition, equivalent to the Kalman rankcondition in control theory, the solutions of these systems decay exponentiallyin time for high frequencies and polynomially for low ones. This allows toderive a sharp description of the space-time decay of solutions for large time.However, such analysis relies heavily on the use of the Fourier transform thatwe avoid here, developing the "physical space version" of the hyperbolichypocoercivity approach introduced by Beauchard and Zuazua, to prove newasymptotic results in the linear and nonlinear settings. The new physical spaceversion of the hyperbolic hypocoercivity approach allows to recover the naturalheat-like time-decay of solutions under sharp rank conditions, withoutemploying Fourier analysis or $L^1$ assumptions on the initial data. Takingadvantage of this Fourier-free framework, we establish new enhanced time-decayestimates for initial data belonging to weighted Sobolev spaces. These resultsare then applied to the nonlinear compressible Euler equations with lineardamping. We also prove the logarithmic stability of the nonlinearly damped$p$-system.

Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: application to the nonlinearly damped p-system