The pointwise estimates to solutions for 1-dimensional linear thermoelastic system with second sound

Abstract

In this paper, we study the thermoelastic system with second sound in one space variable. The mathematical model is a hyperbolic partial differential system with damping term which brings about some parabolic properties. The pointwise estimates of the solution are obtained and from which the generalized Huygens' principle is shown. Our approach is based on detailed analysis on the Green function of the system. Mathematical model and main resultIn this paper, we study the following 1-dimensional thermoelastic system with second sound:where u and θ are the displacement and temperature difference of the elastic media, respectively, q denotes the heat flux, κ, τ, α, β, γ are positive constants.

Nonlinear Well-Posedness and Rates of Decay in Thermoelasticity With Second Sound

“…Here we shall give polynomial decay rates for the solution. This will be based on the knowledge of the asymptotic behavior of solutions to the corresponding linearized system which we can take from the work of Yang & Wang [23] and of W. Wang & Z. Wang [21]. Then suitable a priori estimates have to be proved in the spirit of the method described in general in [14], and already used in classical (hyperbolic-parabolic) thermoelasticity now connected to Cattaneo's law.…”

“…Having given the global solution for small data (u x , u t , θ, q)(t = 0) in H 2 from Tarabek [19], we shall describe in Section 4 the asymptotic behavior of V (t) ≡ (u x , u t , θ, q)(t) as time t → ∞. For the corresponding linearized system Yang and Wang [24], also W. Wang and Z. Wang [21], gave decay rates for L q -norms of V (t) (cp. also Wang and Yang [23] in three space dimensions).…”

Exponential stability for nonlinear thermoelastic equations with second sound

“…His work generalized an earlier one in References [6][7][8] to thermoelasticity with second sound. The Cauchy problem of the linear thermoelastic system with second sound was also studied by Wang and Wang [9] and Yang and Wang [10]. In particular…”