Global solutions to thermomechanical equations with nonconvex Landau- Ginzberg free energy (Q1124406)

The author tuckles boundary and initial value problems (in one space dimension) for a ``thermoelastic'' body, the free energy F of which depends not only on temperature \(\theta\) and linear strain \(\epsilon\) but also on strain gradient: \(F=F_ 0(\theta)+\alpha_ 1(\theta -{\bar \theta})\epsilon^ 2+F_ 3(\epsilon)+(\gamma /2)| \text{grad} \epsilon |^ 2\). Notice the critical temperature \(\theta\) at which the quadratic term changes sign; \(\alpha\) and \(\gamma\) are positive constants: \(F_ 3\) grows at least as \(\epsilon^ 4\) and \(-F_ 0\) grows at least as \(\theta^ 2\) and is subject to other conditions. In writing the energy balance equation, the Fourier law of heat conduction is assumed. The boundary is supposed to be either free or clamped and thermally insulated. A global existence and uniqueness theorem is proved, combining a local existence theorem with a priori estimates of solutions.