Global Existence and Asymptotic Behavior of Solutions for Quasi-linear Dissipative Plate Equation

TL;DR

This paper proves the global existence and decay of solutions for a multi-dimensional quasi-linear dissipative plate equation, overcoming regularity-loss challenges with a novel energy method and providing explicit asymptotic approximations.

Contribution

It introduces a time-weighted energy approach to establish global solutions and decay rates for the dissipative plate equation with regularity-loss properties.

Findings

Established global existence of solutions under small initial data.

Derived optimal decay estimates for lower-order derivatives.

Provided explicit asymptotic approximation of solutions using fundamental solutions.

Abstract

In this paper we focus on the initial value problem for quasi-linear dissipative 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 estimates of solutions. We overcome this difficulty by employing a time-weighted $L^2$ energy method which makes use of the integrability of $\|(\p^2_xu_t,\p^3_xu)(t)\|_{L^{\infty}}$. 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 to prove the global existence and asymptotic decay 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 given explicitly in terms of the fundamental solution of a fourth-order linear parabolic equation.