Existence theory of the nonlinear plate equations

Abstract

This paper is devoted to the theoretical analysis of the nonlinear plate equations in ${\mathbb{R}}^n\times (0,\infty ),$ $n\ge 1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild solutions for small initial data in $L^1({\mathbb{R}}^n)\cap H^s({\mathbb{R}}^n)$-spaces. We also prove the existence and uniqueness of local and global solutions in the framework of Bessel-potential spaces $H_p^s({\mathbb{R}}^n)=(I-\Delta {)}^{s/2}{L}^p({\mathbb{R}}^n).$ In order to derive the existence results we develop new time decay estimates of the solution of the corresponding linear problem.

21 pages