Preprint

Inserted: 31 oct 2026

Year: 2026

Abstract:

We study the decay properties of solutions to the Korteweg–de Vries equation on a $\mathcal{Y}$-junction. First, we identify suitable boundary conditions that ensure the problem is globally well-posed in the space $L^2(\mathcal{Y})$. Then, using virial inequalities in the spirit of Muñoz and Ponce (2019), we establish decay, in the sense of $\liminf$, of the $L^2$ norm of the solution in certain time-dependent regions far from the solitons. We consider both weakly and strongly dominated linear regions. These asymptotic properties require relations between the boundary condition parameters and the nonlinearity coefficients. Finally, we discuss adaptations to general star graphs.