In this work, we investigate the initial-boundary value problem (IBVP) of the Cahn–Hilliard-type model on the half-line, model related to heat-mass transfer phenomena and solid fluid dynamics. Recently, Chatziafratis et al. (Math. Models Methods Appl. Sci. 35 (2025) 1133-1197) employed the Fokas method to solve the IBVP for Cahn–Hilliard-type model on the half-line and derived properties such as regularity near the boundary of the domain. Inspired by this work, we rigorously prove the well-posedness in the sense of Hadamard with initial condition \(u_{0}(x) \in H^{s}(0,\infty )\) , for regularity exponent \(\frac{1}{2}<s<\frac{9}{2}\) and \(s \ne \frac{3}{2}\) under Dirichlet–Neumann boundary conditions. To effectively overcome the difficulties posed by nonlinear term and boundary conditions, the core of this work is to investigate the associated forced linear problem, which can be solved by the Fokas method. Given the complex characteristics of the linear high-order terms, we utilize analytical techniques such as the \(L^{2}\) -boundedness of the Laplace transform and the properties of Banach algebras to establish linear estimates, laying foundation for the proof of well-posedness. Furthermore, this work identifies the appropriate function spaces for the Dirichlet and Neumann boundary data, which is crucial for the overall analysis.
