Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite dimensional Hilbert space

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of solutions to a class of nonlinear vector equations in finite-dimensional Hilbert spaces, providing conditions under which solutions are well-defined and globally stable.

Contribution

It establishes conditions for existence, uniqueness, and global behavior of solutions to a specific nonlinear vector equation, extending previous results to more general settings.

Findings

Proved existence and uniqueness of solutions under certain assumptions.

Analyzed the global behavior and stability of solutions.

Provided conditions ensuring solutions do not blow up in finite time.

Abstract

The initial value problem and global properties of solutions are studied for the vector equation: $\Big(\|u'\|^{l}u'\Big)'+\|A^{\frac{1}{2}}u\|^\beta Au+g(u')=0$ in a finite dimensional Hilbert space under suitable assumptions on $g$.