ABSTRACT
This chapter establishes the results on decay rates of solutions to both initial value problems and initial boundary value problems for linear parabolic equations and two classes of linear hyperbolic-parabolic coupled systems: linear one-dimensional thermoelastic systems and thermoviscoelastic systems. It shows the obtaining of the results on decay rates of solution to the initial value problem for the heat equation by using the Poisson formula and the Young inequality. It turns out that the decay rates depend on the space dimension n. The chapter also looks at the initial boundary value problem for linear parabolic equations. It examines the initial value problems for linear one-dimensional thermoelastic systems and thermoviscoelastic systems as well as the initial boundary value problems for both linear thermoelastic and thermoviscoelastic systems. The chapter also establishes a theorem to show that if the infinitesimal generator is invertible, then the exponential stability of the Co-semigroup is equivalent to the exponential decay of the solution.
