A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks

Abstract

In an open bounded interval $\hat{I}\text{c◯}$, the problem \[ u_{tt} = u_{xx} - \big(f(Θ)\big)_x, Θ_t = Θ_{xx} - f(Θ) u_{xt}, \] is considered under the boundary conditions $u{|}_{\partial \hat{I}\text{c◯}}=\hat{I}{˜}_x{|}_{\partial \hat{I}\text{c◯}}=0$, and for $f\in C^2([0,\infty ))$ satisfying $f(0)=0$, $f^{\prime }>0$ on $[0,\infty )$ and $f^{\prime }\in W^{1,\infty }((0,\infty ))$. In the sense of unconditional global existence, uniqueness and continuous dependence, this problem is shown to be well-posed within ranges of initial data merely satisfying \[ u_0\in W_0^{1,2}(Ω), \quad u_{0t} \in L^2(Ω) \quad \mbox{and} \quad Θ_0 \in L^2(Ω) \mbox{ with $\hat{I}˜\ge 0$ a.e.~in $\hat{I}\text{c◯}$,} \] and in classes of solutions fulfilling \[ u\in C^0([0,\infty);W_0^{1,2}(Ω)), \qquad u_t \in C^0([0,\infty);L^2(Ω)) \qquad \mbox{and} \qquad Θ\in C^0([0,\infty);L^2(Ω)) \cap L^2_{loc}([0,\infty);W^{1,2}(Ω)). \]