A decreasing property of solutions of parabolic equations with applications to thermoelasticity
W. A. Day · Quarterly of Applied Mathematics · 1983
Introduction.The object of this paper is to establish a decreasing property of solutions of parabolic equations which satisfy boundary conditions of a somewhat unusual kind.We shall begin by illustrating, with the aid of two examples from the quasi-static theory of thermoelasticity, how such problems can arise.Let us consider first the coupled partial differential equations 820 80 82u82u 80 d? ' ^ + 2"> Tx which describe the behavior of a slab -l<x<l made of homogeneous and isotropic material.Here 9(x, t) is the temperature, u(x, t) is the displacement component in the direction of the x-axis, 0o is a uniform reference temperature, k is the conductivity, c is the specific heat at constant strain, a is the coefficient of expansion, and A, ^ are the elastic moduli.The reader is referred to Carlson's article [1] or to Boley and Weiner's treatise [2] for derivations of these equations.We shall suppose the faces of the slab to be maintained at the reference temperature and to be clamped, that is to say 6(-l, t) = 0(1, t) = 0o, u(-l, t) = u(l, t) = 0.We can write the differential equations as an energy equation ke-^ = o *5 dx2 0 dt and a quasi-static equation of motion 8 a/ox ' 0, respectively, where ri = (0 -60) + a(3A + 2n) Ĉ/q OX