Spatial decay estimates for a coupled system of second-order quasilinear partial differential equations arising in thermoelastic finite anti-plane shear (Q1372385)

The authors investigate the spatial decay behavior of solutions of a coupled system of second-order quasilinear partial differential equations in divergence form, \[ [M(q, \theta)u_{,\beta}]_{,\beta}= 0,\qquad [K(q, \theta)\theta_{,\beta}]_{,\beta}= 0, \] defined on a two-dimensional semi-infinite strip. Here \(q =|\nabla u|,\) \(u\) and \(\theta\) are the displacement and temperature fields, respectively, and the functions \(M,K\) are constitutive functions of their indicated arguments. Such equations arise in the theory of anti-plane shear deformations for compressible isotropic nonlinear thermoelastic materials. For a review of anti-plane shear see \textit{C. O. Horgan} [SIAM Rev. 37, 53-81 (1995; Zbl 0824.73018)]. The ``energy'' method using differential inequalities for quadratic integrals is employed to obtain exponential decay estimates, which are illustrated by some examples. The results are relevant to Saint-Venant principles for nonlinear thermoelasticity as well as to theorems of Phragmen-Lindelöf type.