Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations

In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation

$${}_0^C{D}_t^{\beta }u+(-\mathrm{\Delta }{)}^{\sigma }u+(-\mathrm{\Delta }{)}^{\sigma }{}_0^C{D}_t^{\beta }u=-(-\mathrm{\Delta }{)}^{\sigma }f(u)$$

with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $(-\mathrm{\Delta }{)}^{\sigma }$. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $N$ by Banach fixed point theorem.