Higher-order diffusion and Cahn-Hilliard-type models revisited on the half-line

TL;DR

This paper rigorously analyzes higher-order diffusion and Cahn-Hilliard models on the half-line, providing explicit solutions, regularity, asymptotic behavior, and counterexamples, advancing the understanding of complex PDEs in applied sciences.

Contribution

It introduces explicit solution representations and rigorous analysis for fourth-order PDE models on the half-line, extending the Fokas method to higher-order equations and exploring their qualitative properties.

Findings

Derived effective solution formulas for higher-order diffusion models.

Proved convergence, regularity, and asymptotic properties of solutions.

Constructed a counter-example illustrating non-uniqueness.

Abstract

In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn-Hilliard (C-H) model (also Kuramoto-Sivashinsky equation), formulated in the spatiotemporal quarter-plane. Such models are of relevance to heat-mass transfer phenomena, solid-fluid dynamics and the applied sciences. In particular, we derive formally effective solution representations, justifying a posteriori their validity. This includes the reconstruction of the prescribed initial and boundary data, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula is utilized to rigorously deduce the solution's regularity and asymptotic properties near the boundaries of the domain, including uniform convergence, eventual (long-time) periodicity under (eventually) periodic boundary conditions, and null non-controllability. Importantly, this analysis is indispensable for exploring the (non)uniqueness of the problem's solution and a new counter-example is constructed. Our work is based on the synergy between: (i) the well-known Fokas unified transform method and (ii) a new approach recently introduced for the rigorous analysis of the Fokas method and for investigating qualitative properties of linear evolution partial differential equations (PDE) on semi-infinite strips. Since only up to third-order evolution PDE have been investigated within this novel framework to date, we present our analysis and results in an illustrative manner and in order of progressively greater complexity, for the convenience of readers. The solution formulae established herein are expected to find utility in well-posedness studies for nonlinear counterparts too.