The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time of the solution is independent of the spatial variable.