Global Existence and Decay of Solutions for a Quasi-linear Dissipative Plate Equation

In this paper we focus on the initial value problem of a quasi-linear dissipative plate equation with arbitrary spatial dimensions (n ≥ 1). This equation verifies the decay property of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted (with negative exponent) L2 energy method combined with the optimal L2 decay estimates of lower-order derivatives of solutions. We obtain the global existence and optimal decay estimates of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is the fundamental solution of the corresponding fourth-order linear parabolic equation.