Remarks on propagation of singularities in thermoelasticity (Q1604463)

The following one-dimensional displacement-temperature initial-boundary value problem of nonlinear thermoelasticity is studied: find a pair \((u,\theta)\) that satisfies the field equations \[ \begin{gathered} u_{tt}- \alpha^2 u_{xx}+ \gamma_1\theta_x= f(u,\theta),\\ \theta_t- \beta^2\theta_{xx}+ \gamma_2 u_{tx}= g(u,\theta)\quad\text{for }| x|< \infty,\;t> 0\end{gathered}\tag{1} \] subject to the initial conditions \[ u(0,x)= u_0(x),\;u_t(0,x)= u_1(x),\;\theta(0,x)= \theta_0(x)\quad \text{for }| x|<\infty.\tag{2} \] In (1) \(\alpha^2\), \(\beta^2\), \(\gamma_1\), \(\gamma_2\) are prescribed functions of \(x\) and \(t\); and \(f\) and \(g\) are prescribed nonlinear functions of \(u\) and \(\theta\); and in (2) \(u_0\), \(u_1\), \(\theta_0\) are prescribed functions of \(x\). Five theorems on propagation of singularities for the problem (1)--(2) are proved. In particular, it is shown that for the linear problem \((f= g= 0)\) when \(u_0\) and \(u_1\) are smooth, a non-smooth function \(\theta_0\) will produce singularities for \(u\) when \(t> 0\). For the nonlinear problem, a concept of microlocal singularities is introduced, and it is shown that such singularities propagate along bi-characteristic strips of the hyperbolic operator \(\partial^2_t- \alpha^2\partial^2_x\).