Continuum Mech. Thermodyn. DOI 10.1007/s00161-017-0556-z ORIGINAL ARTICLE Wenjun Liu · Miaomiao Chen Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback Received: 12 November 2016 / Accepted: 25 January 2017 © Springer-Verlag Berlin Heidelberg 2017 Abstract In this paper, we study the well-posedness and exponential decay for the porous thermoelastic system with the heat conduction given by Cattaneo’s law and a time-varying delay term, the coefficient of which is not necessarily positive. Using the semigroup arguments and variable norm technique of Kato, we first prove that the system is well-posed under a certain condition on the weight of the delay term, the weight of the elastic damping term and the speed of the delay function. By introducing a suitable energy and an appropriate Lyapunov functional, we then establish an exponential decay rate result. Keywords Porous thermoelastic system · Elastic damping · Time-varying delay · Decay 1 Introduction In this paper, we investigate the well-posedness and decay properties of solutions for the following porous thermoelastic system coupled with the heat conduction by Cattaneo’s law, with an elastic damping term and a time-varying delay term in the internal feedback ⎧ ⎪ ⎨ ⎪ ⎩ ρ u tt = μu xx + bφ x − γ 1 u t − γ 2 u t (x , t − τ(t )), (x , t ) ∈ (0, 1) × (0, ∞), J φ tt = αφ xx − bu x − ξφ + βθ x , (x , t ) ∈ (0, 1) × (0, ∞), cθ t =−q x + βφ tx − δθ, (x , t ) ∈ (0, 1) × (0, ∞), τ 0 q t + q + k θ x = 0, (x , t ) ∈ (0, 1) × (0, ∞), (1.1) with initial data and boundary conditions ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u (x , 0) = u 0 (x ), φ(x , 0) = φ 0 (x ), θ(x , 0) = θ 0 (x ), x ∈[0, 1], q (x , 0) = q 0 , u t (x , 0) = u 1 (x ), φ t (x , 0) = φ 1 (x ), x ∈[0, 1], u (0, t ) = φ x (0, t ) = θ(0, t ) = 0, t ∈[0, ∞), u (1, t ) = φ x (1, t ) = θ(1, t ) = 0, t ∈[0, ∞), u t (x , t − τ(0)) = f 0 (x , t − τ(0)), (x , t ) ∈ (0, 1) ×[0,τ(0)), (1.2) where u is the longitudinal displacement, φ is the volume fraction difference, θ is the temperature difference, q is the heat flux, γ 2 is real number and ρ,μ, b,γ 1 , J , α, b,ξ,β, c,δ,τ 0 , k are positive constants with b,μ,ξ satisfying μξ > b 2 ,γ 2 is a constant not necessarily positive, and τ(t )> 0 represents the time-varying delay. System (1.1)–(1.2) arises in the theory of linear elastic materials with voids or vacuous pores, which was established by Cowin and Nunziato [7, 8, 31]. In recent years, elastic materials with voids, which have nice Communicated by Andreas Öchsner. W. Liu (B ) · M. Chen College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China E-mail: wjliu@nuist.edu.cn