A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs
This paper investigates an initial-boundary value problem for the Korteweg--de Vries (KdV) equation on a finite equilateral star graph. We use the notion of $s$-compatibility, which extends the classical compatibility conditions to the coupled boundary conditions arising at the central vertex, inspired by the work of Bona, Sun, and Zhang [14]. By combining analytical estimates, fixed-point arguments, and nonlinear interpolation on suitable augmented spaces, we establish global well-posedness for the nonlinear problem in $H^s$, $0\leq s\leq3$. At the critical indices $s=1/2,3/2,5/2$, the admissible data are defined through constrained interpolation; away from these indices, they are characterized by the corresponding pointwise compatibility conditions. The nonlinear compatibility condition at the central vertex is handled by introducing an auxiliary variable, which allows the interpolation argument to be carried out on a pair of linear spaces.