On the decay rates for thermoviscoelastic systems of type III Muhammad I. Mustafa King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, P.O. Box 860, Dhahran 31261, Saudi Arabia article info Keywords: Thermoelasticity of type III Viscoelastic damping General decay Convexity abstract In this paper we consider an n-dimensional thermoelastic system of type III with viscoelas- tic damping. We establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a larger class of relaxation functions and generalizes previous results existing in the literature. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction In this paper we are concerned with the following problem u tt  lDu ðl þ kÞrðdiv uÞþ R t 0 gðt  sÞDuðsÞds þ br# ¼ 0; in X ð0; 1Þ b# tt  kD# þ bdiv u tt  cD# t ¼ 0; in X ð0; 1Þ uðx; tÞ¼ #ðx; tÞ¼ 0; on @X ð0; 1Þ uðx; 0Þ¼ u 0 ðxÞ; u t ðx; 0Þ¼ u 1 ðxÞ; #ðx; 0Þ¼ # 0 ðxÞ; # t ðx; 0Þ¼ # 1 ðxÞ; x 2 X 8 > > > < > > > : ð1:1Þ a thermoviscoelastic system of type III associated with homogeneous Dirichlet boundary conditions and initial data in suit- able function spaces. Here X is a bounded domain of R n ðn P 2Þ with a smooth boundary @X; u ¼ uðx; tÞ2 R n is the displace- ment vector, # ¼ #ðx; tÞ is the difference temperature, and the relaxation function g is a positive nonincreasing function. The coefficients b; k; b; c; l; k are positive constants, where l; k are Lame moduli. In this work, we study the decay properties of the solutions of (1.1) for functions g of general-type decay. In classical thermoelasticity, the heat conduction is governed by the Fourier’s law, which means that the heat flux is pro- portional to the gradient of temperature. This theory predicts an infinite speed of heat propagation; that is any thermal dis- turbance at one point has an instantaneous effect elsewhere in the body. Experiments showed that heat conduction in some situations is free of this paradox and disturbances, which are almost entirely thermal, propagate in a finite speed. To over- come this physical paradox, Green and Naghdi in [5–7] re-examined the classical model and introduced the so-called models of thermoelasticity of type II and III. The systems arising in thermoelasticity of type III are of dissipative nature whereas those of type II thermoelasticity do not sustain energy dissipation. For a discussion of these new models of thermoelasticity, we refer to [2–4,14,15], and for classical thermoelasticity, we refer to books by Jiang and Racke [8] and Zheng [18]. In the absence of the viscoelastic term, Quintanilla and Racke [16] and Zhang and Zuazua [17] independently studied the decay of energy for the problem of the linear thermoelastic system of type III by using the spectral method and the classical energy method, and they obtained the exponential stability in one space dimension for various types of boundary conditions. In the multi-dimensional case the situation is much different. It was shown that the dissipation given by heat conduction is http://dx.doi.org/10.1016/j.amc.2014.04.042 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. E-mail address: mmustafa@kfupm.edu.sa Applied Mathematics and Computation 239 (2014) 29–37 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc