The Initial‐Boundary Value Problem of Klein–Gordon Equation on the Half‐Line

ABSTRACT This work investigates the initial‐boundary value problem (IBVP) of the Klein–Gordon (KG) equation on the half‐line within the Sobolev spaces framework. By employing the Fokas method coupled with the Banach fixed‐point theorem, we establish the following key results: (i) For the IBVP of linear KG equation, we prove the well‐posedness results through decomposition into a free Cauchy problem and a forced IBVP with homogeneous data. A priori linear estimates for these decomposed problems are rigorously derived. (ii) The IBVP of the nonlinear KG equation is systematically analyzed via the Banach fixed‐point theorem in the space , which establishes local well‐posedness under the regularity condition , . (iii) A synthesis of the Fokas method with Sobolev spaces techniques extends the applicability of the Fokas method to fractional regularity regimes. The methodology provides explicit solution representations while maintaining appropriate regularity matching between initial and boundary data. This work significantly advances the functional framework for IBVP analysis on unbounded domains, bridging modern transform methods with classical Sobolev space theory.