Nonlocal stabilization by starting control of the normal equation generated from Helmholtz system

TL;DR

This paper investigates the stabilization of a semilinear parabolic equation linked to the 3D Helmholtz system using starting control, establishing key inequalities to ensure solutions decay to zero.

Contribution

It introduces new inequalities related to starting control for the normal parabolic equation associated with the Helmholtz system, extending previous results.

Findings

Established two new inequalities for starting control

Proved the effectiveness of control in stabilizing solutions to zero

Extended the theoretical framework for Helmholtz-related parabolic equations

Abstract

We consider the semilinear parabolic equation of normal type connected with the 3D Helmholtz equation with periodic boundary condition. The problem of stabilization to zero of the solution for normal parabolic equation with arbitrary initial condition by starting control is studied. This problem is reduced to establishing three inequalities connected with starting control, one of which has been proved previously. The proof for the other two is given here.