The Periodic Cauchy Problem For The IMBq Equation With Strong Damped Term

In this paper, we consider the periodic Cauchy problem for the IMBq equation with strong damped term utt-uxxtt-uxx-vuxxt=f(u)xx, x∈R, t>0,(0.2) Where u(x+2π, t)=u(x,t), v>0. Firstly, we establish the decay estimates of the solution to the linear IMBq equation. Secondly, we will study the existence and uniqueness of the local solution to the periodic cauchy problem for the IMBq equation under smallness condition on the initial data. Finally, we will study the existence and uniqueness and the decay estimates of the global solution to the equation under smallness condition on the initial data.