Collective dynamics of the confined kinetic Cucker–Smale model with non-compact support

In this paper, we study the kinetic Cucker–Smale (KCS) model with external harmonic potential and non-compactly supported initial data. The presence of an external harmonic potential introduces a confining effect, which competes with the alignment mechanism and leads to large-time behaviour significantly different from that of the original KCS model. We establish an unconditional collective behaviour in which velocities and positions converge towards a harmonic oscillator with different rates. Specifically, the convergence is exponential when β = 0 , but algebraic when 0 < β < 1 , showing a sharp transition in the convergence rate determined by the communication weight parameter β . By introducing a carefully designed Lyapunov fluctuation functional, modified position-velocity moments, and a time-dependent position-velocity space splitting method, we are able to characterize the emergence of collective dynamics determined by initial distributions.