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Initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity are considered, and the existence of global classical solutions is established by means of the Leray-Schauder fixed point theorem. Introduction. In this paper we study the existence of global smooth solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity. The conservation laws of mass, momentum, and energy for one-dimensional materials with the reference density p0 = 1 are u,-vx = 0, Vt ~ °x = 0 ' / 2X «. (1.2) where subscripts indicate partial differentiations, u is the deformation gradient, v is the velocity, e denotes the internal energy, a is the stress, rj stands for the specific entropy, 6 for the temperature, and q for the heat flux. For one-dimensional, homogeneous, thermoviscoelastic materials, e, a , rj, and q are given by the constitutive relations (see [1]) e = e(u,6), a = a(u, 6, vx), ri = r)(u,6), q = q{u,6,0x), (1.3) which in order to be consistent with (1.2), must satisfy a(u, 6, 0) = \j/u{u, 6), fj(u, 0) = -&e(u, 0), (a(u, 0,w)-d(u,9, 0))w >0, q(u, 9, g)g < 0, where y/ — e 9t] is the Helmholtz free energy function. (1.4) Received August 5, 1991 and, in revised form, October 28, 1991. 1991 Mathematics Subject Classification. Primary 35M10, 73C35, 73B30. Permanent address : Department of Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi Province, PR China. E-mail address : unm204@ibm.rhrz.uni-bonn.de. © 1993 Brown University 731

Global large solutions to initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity