Existence theory of the nonlinear plate equations

TL;DR

This paper establishes the existence and uniqueness of solutions for nonlinear plate equations in various function spaces, using new decay estimates for the linear problem, advancing the theoretical understanding of such nonlinear PDEs.

Contribution

It provides the first comprehensive existence and uniqueness results for nonlinear plate equations with polynomial nonlinearity in multiple functional frameworks.

Findings

Proved global mild solutions for small initial data in $L^{1} igcap H^s$ spaces.

Established local and global solutions in Bessel-potential spaces $H^s_p$.

Developed new decay estimates for the linear problem's solutions.

Abstract

This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ 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 $H^s_p(\mathbb{R}^n)=(I-\Delta)^{s/2}L^p(\mathbb{R}^n).$ In order to derive the existence results we develop new time decay estimates of the solution of the corresponding linear problem.