Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity

Abstract

FOR THE equations of one-dimensional nonlinear thermoelasticity Slemrod [17] established in 1981 the global existence of smooth solutions for small initial data in the case of a bounded reference configuration for a homogeneous medium. The boundary conditions (traction free, constant temperature or rigidly clamped, insulated) he considered yield, through the differential equation, an additional boundary condition which was crucial for obtaining the necessary energy estimates. Subsequently, Jiang [7] discussed the same boundary conditions as in [17] in the case of an unbounded reference configuration (the half line). Kawashima [lo], Zheng and Shen [ 181, and Hrusa and Tarabek [6] dealt with the one-dimensional Cauchy problem. Recently, Racke and Shibata [13] proved a global existence theorem for the Dirichlet initial boundary value problem (rigidly clamped, constant temperature) in a bounded interval by using the decay estimates for the linearized equations and the L*-energy method, and the author [8] obtained a similar result for the problem in an unbounded interval (the half line) by only applying the L*-energy method. More recently, using a method similar to that in [13], Shibata [16] showed that global small solutions exist for the Neumann problem (traction free, insulated) in the case of a bounded reference configuration. For an unbounded reference configuration (the half line), the case of the Neumann boundary conditions remained open. Our purpose in this paper is to prove a global existence theorem for the Neumann problem in the half line and in a bounded interval for small data by applying the L2-energy method. The proof given here closely follows the pattern of the proof in [8], subject to the necessary modifications. We note that a globally defined classical solution should not be expected for large data. Indeed, Dafermos and Hsiao [4], and Hrusa and Messaoudi [5] showed that for specialized constitutive equations if the initial data are large, then the solution to the Cauchy problem will blow up in finite time. For three-dimensional nonlinear models, we refer the reader to the papers [3, 9, 10-12, 151 (and the references cited therein). This paper is organized as follows. In Section 1 the equations of one-dimensional nonlinear thermoelasticity are presented and the main theorem is stated. In Section 2 we give a local