Global Well-Posedness for the 3D Damped Benjamin–Bona–Mahony Equation in $$L^{p}$$ Based Sobolev Space

The global well-posedness of the damped Benjamin-Bona-Mahony equation is investigated in the $$L^{p}$$-based Sobolev space class $$\dot{H}^{s,p}({\mathbb {T}}^{3}).$$ Firstly, using bilinear estimates and contraction mapping principle, the local well-posedness of the equation is originally shown in $$\dot{H}^{s,p}({\mathbb {T}}^{3})$$ with $$s \ge \max \{0,\frac{3}{p}-1\}$$ and $$\frac{3}{2}\le p<\infty .$$ Secondly, using high-low frequency decomposition and Sobolev embedding, the global well-posedness of the damped Benjamin–Bona–Mahony is furthermore studied in $$\dot{H}^{s,p}({\mathbb {T}}^{3})$$ with $$ \max \{0,\frac{3}{p}-1\}\le s\le 1$$ and $$\frac{3}{2}\le p\le 2$$.