Asymptotic behavior of solutions to the system of one-dimensional nonlinear thermoviscoelasticity
The authors study the large time behavior of solutions to the system of equations modeling one-dimensional thermoviscoelasticity: \(u_t=v_x\), \(v_t=\sigma_x\), \((e+v^2/2)_t=(\sigma v)_x -q_x\). The system is supplemented with the constitutive relations \(e=c_0 \theta\), \(\sigma= - f(u)\theta +\mu(u)v_x\), \(q=-k(u,\theta)\theta_x\) and solutions are subject to initial and Dirichlet boundary conditions. The main result of the paper is to show that globally defined smooth solutions may possess phase transitions when the material function \(f\) is not monotone. In the case \(f\) is strictly decreasing it is furthermore shown that \(v(.,t)\to 0\), \(\theta(.,t)\to T_0\) and \(u(.,t)\to \widetilde z\) as \(t\to\infty\) in the space \(H^1(0,1)\), i.e. any solution tends to a stable state.
scientific article; zbMATH DE number 1159353