L^{2}-blowup estimates of the plate equation

TL;DR

This paper investigates the long-term behavior of solutions to the n-dimensional plate equation with weighted initial data, revealing conditions under which solutions exhibit infinite time blowup, especially when the initial velocity's zero-th moment is non-zero.

Contribution

It provides optimal L^{2}-norm estimates for solutions of the plate equation in dimensions 1 to 4, highlighting blowup phenomena related to initial data moments.

Findings

Solutions blow up over infinite time when initial velocity's zero-th moment is non-zero.

Derived optimal estimates for the L^{2}-norm in dimensions 1 to 4.

Identified conditions leading to infinite time blowup in the plate equation.

Abstract

We consider the Cauchy problems in n-dimensional Euclidean space for the plate equation with a weighted L^{1}-initial data. We derive optimal estimates of the L^{2}-norm of solutions for n = 1, 2, 3, 4. In particular, such obtained results express infinite time blowup properties in the case when the 0-th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from an already developed technique by the author's collaborative works.