Global existence and asymptotic behaviorof solutions for quasi-linear dissipative plate equation

In this paper we focus on the initial value problem for quasi-lineardissipative plate equation in multi-dimensional space $(n\geq2)$.This equation verifies the decay property of the regularity-loss type,which causes the difficulty in deriving the global a priori estimatesof solutions. We overcome this difficulty by employing atime-weighted $L^2$ energy method which makes use ofthe integrability of $||$(∂$^2_xu_t,$∂$^3_xu)(t)||_{L^{\infty}}$.This $L^\infty$ norm can be controlled by showing the optimal $L^2$decay estimates for lower-order derivatives of solutions.Thus we obtain the desired a priori estimate which enables us toprove the global existence and asymptotic decay of solutionsunder smallness and enough regularity assumptions on the initial data.Moreover, we show that the solution can be approximated bya simple-looking function, which is given explicitly in terms ofthe fundamental solution of a fourth-order linear parabolic equation.