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This article focuses on the Cauchy problem of the Benjamin–Bona–Mahony–Burgers (BBMB) equation with nonlocal control constraints inℝ3{\mathbb{R}^{3}}. We employ the Phragmén–Lindelöf decomposition method and Green’s function to investigate global classical solutions and their long-term behaviors, including optimal estimates for finite density initial perturbation. Additionally, optimal solutions near the minimal mass ground state are analyzed.

Global classical solution of the Cauchy problem to the 3D Benjamin–Bona–Mahony–Burgers-type equation with nonlocal control constraints