Acta Mathematica Scientia 2009,29B(3):645–649 http://actams.wipm.ac.cn SINGULAR LIMITS FOR INHOMOGENEOUS EQUATIONS OF ELASTICITY * Dedicated to Professor Wu Wenjun on the occasion of his 90th birthday Lu Yunguang ( ) Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Department of Mathematics, National University of Colombia, Bogota, Colombia E-mail: yglu 2000@yahoo.com Christian Klingenberg Mathematisches Institut, Universit¨at W¨ urzburg, Am Hubland, 97074 W¨ urzburg, Germany E-mail: klingenberg@mathematik.uni-wuerzburg.de Abstract Based on the framework introduced in [4] or [5], the singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of inhomogeneous equations of elasticity is studied. We are able to reach equilibrium even though the nonlinear stress term is not strictly increasing. Key words relaxation limit; equations of elasticity; compensated compactness; invariant regions theory 2000 MR Subject Classification 35L65; 35B40 1 Introduction In this paper, we will study the singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of inhomogeneous equations of elasticity with relaxation and diffusion      v t - u x + g(v,u)= εv xx , u t - s(v) x + f (v,u)+ u - h(v) τ = εu xx , (1.1) with bounded initial data (v(x, 0),u(x, 0)) = (v 0 (x),u 0 (x)), (1.2) where v denotes the strain, the nonlinear function s(v) is the stress and u the velocity. The second equation in (1.1) contains a relaxation mechanism with h(v) as the equilibrium value for u, τ the relaxation time and ε is the diffusion coefficient. In the literature relaxation limits without coupled viscosity have been studied. Noteworthy references are Chen, Levermore and Liu [2] and Natalini [6] where the relaxation limit for (1.1) * Received December 4, 2008