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

In an open bounded interval $\mathrm{\Omega }$, the problem $$u_{tt}=u_{xx}-(f(\mathrm{\Theta }){)}_x,{\mathrm{\Theta }}_t={\mathrm{\Theta }}_{xx}-f(\mathrm{\Theta })u_{xt},$$ is considered under the boundary conditions $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}={\mathrm{\Theta }}_x{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$, and for $f\in C^2([0,\mathrm{\infty }))$ satisfying $f(0)=0$, $f^{\mathrm{\prime }}>0$ on $[0,\mathrm{\infty })$ and $f^{\mathrm{\prime }}\in W^{1,\mathrm{\infty }}((0,\mathrm{\infty }))$. 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 and \quad Θ_0 \in L^2(Ω) with $Θ\ge 0$ a.e.~in $Ω$, and in classes of solutions fulfilling $$u\in C^0([0,\mathrm{\infty });W_0^{1,2}(\mathrm{\Omega })),u_t\in C^0([0,\mathrm{\infty });L^2(\mathrm{\Omega }))and\mathrm{\Theta }\in C^0([0,\mathrm{\infty });L^2(\mathrm{\Omega }))\cap L_{loc}^2([0,\mathrm{\infty });W^{1,2}(\mathrm{\Omega }))\mathrm{.}$$