Fractional Klein–Gordon equations in orthants: well-posedness and scattering

Abstract

We study the initial–boundary value problem for a fractional nonlinear Klein–Gordon equation posed on the hyperoctant Dn,n⩾2 with nonhomogeneous boundary data on each coordinate hyperplane. The equation involves a spatial fractional Laplacian of Caputo type with order β∈(3/2,2), which induces strong nonlocal dissipation and smoothing effects. Using multidimensional Laplace transform techniques, we develop a Hilbert–Riemann–type framework that yields an explicit integral representation for solutions of the associated linear problem, including a boundary operator that captures the influence of boundary forcing without imposing compatibility conditions. Based on this representation, we establish local well-posedness for the nonlinear problem in low-regular Sobolev spaces. For spatial dimensions n=2,3, we further prove global well-posedness for small initial and boundary data and derive sharp large-time asymptotics. In particular, we show that if the boundary data decay polynomially in time, then the solution decays at the same rate and scatters to an explicit asymptotic profile determined by the boundary forcing. This reveals a boundary-driven scattering mechanism that has no analogue in the Cauchy problem and highlights the dominant role of boundary effects in fractional dynamics.