On regular solutions to two-dimensional thermoviscoelasticity

A two-dimensional thermoviscoelastic system of Kelvin–Voigt type with strong dependence on temperature is considered. The existence and uniqueness of a global regular solution is proved without small data assumptions. The global existence is proved in two steps. First, a global a priori estimate is derived by applying anisotropic Sobolev spaces with a mixed norm. Then local existence, proved by the method of successive approximations for a sufficiently small time interval, is extended step by step in time. By a two-dimensional solution we mean that all the relevant quantities depend on two space variables only.