Optimal decay rates and small global solutions to the dissipative Boussinesq equation

This work is devoted to the small amplitude solutions for the initial value problem of the multidimensional dissipative Boussinesq equation. We firstly derive the pointwise estimates of the fundamental solutions by the energy method in the Fourier space. We give the asymptotic profiles of solutions to the corresponding linear problem to get the optimal decay rate for the -norm of solutions in all space dimensions. Under smallness assumptions on the initial data, we study the global existence and uniqueness of solutions by the contractive mapping principle in the solution spaces with time weighted norm.