Local solvability with temperature-dependent gradient coefficient

Establish local-in-time solvability for the temperature-dependent Cahn–Hilliard system (equation (EICHT)) when the gradient-energy coefficient \(\varepsilon\) depends on the temperature, even when the coefficients \(\gamma\), \(\kappa\), and \(d\) are positive constants.

Background

The paper studies temperature-dependent Cahn–Hilliard equations derived from thermodynamic principles. Its local well-posedness theory applies when the gradient-energy coefficient $\varepsilon$ is independent of temperature, equivalently when $\eta_{\nabla w}=0$. If $\varepsilon$ depends on temperature, the temperature equation contains the higher-order term $\eta_{\nabla w}\cdot\nabla\operatorname{div}j$, and the authors’ maximal-regularity framework does not cover the resulting system.

The unresolved problem is to prove local-in-time existence and uniqueness for this more general temperature-dependent coefficient case, including the constant-positive-coefficient setting $\gamma,\kappa,d>0$.