Global Existence and Decay Property for the Cauchy Problem of the Nonlinear MGT Plate Equation

We study the asymptotic behavior of the nonlinear MGT plate equation in the unbounded domain. By using semigroup theory, we first establish the well-posedness result for the Cauchy problem related to the linear MGT plate equation. By using the energy method in the Fourier space, we then prove the optimal decay estimate results for the non-critical case, in which the optimality is analyzed by considering the asymptotic expansion of the eigenvalues. By using the contraction mapping, we also show the local existence for the Cauchy problem of the nonlinear plate in appropriate function spaces, based on which we prove a global existence result for small data by using a priori energy estimates. Finally, based on the decay estimation of linear problems, the decay results of nonlinear problems are obtained.