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nonlinear and degenerating PDE system; global existence; uniqueness; long-time behavior of solutions; $\omega $-limit; phase transitions; thermoviscoelastic materials
This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a {\it strongly nonlinear} internal energy balance equation, governing the evolution of the absolute temperature $\vartheta $, an evolution equation for the phase change parameter $\chi $, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable ${\bf u}$. The main novelty of the model is that the equations for $\chi $ and ${\bf u}$ are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for ${\bf u}$ which needs to be carefully handled. \endgraf First, we prove a global well-posedness result for the related initial-boundary value problem. Secondly, we address the long-time behavior of the solutions in a simplified situation. We prove that the $\omega $-limit set of the solution trajectories is nonempty, connected and compact in a suitable topology, and that its elements solve the steady state system associated with the evolution problem.
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