Rate of convergence to equilibrium for the heated string

Abstract

In the present manuscript, we calculate the exponential rate of convergence of the heated string system (a mixed-type hyperbolic-parabolic system of PDEs) towards the equilibrium, independently of the initial data. As a by-product of our analysis, we obtain an enhanced time decay of the solution. The main tool of our reasoning consists of asymptotic analysis at the Fourier side of the linearized problem in the spirit of Kato. The latter method fits very well to the estimates obtained earlier for the system. The matching between the linear problem and the nonlinear original system requires delicate estimates of the nonlinear terms at the Fourier side as well as careful spectral analys