  In this paper, we study the nonlinear dissipative Boussinesq equation in the
whole space $\mathbb{R}^n$ with $L^1$ integrable data. As our preparations, the
optimal estimates as well as the optimal leading terms for the linearized model
are derived by performing the WKB analysis and the Fourier analysis. Then,
under some conditions on the power $p$ of nonlinearity, we demonstrate global
(in time) existence of small data Sobolev solutions with different regularities
to the nonlinear model by applying some fractional order interpolations, where
the optimal growth ($n=2$) and decay ($n\geqslant 3$) estimates of solutions
for large time are given. Simultaneously, we get a new large time asymptotic
profile of global (in time) solutions. These results imply some influence of
dispersion and dissipation on qualitative properties of solution.
