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

The initial value problem and global properties of solutions are studied for the vector equation: $\Big(|u’|^{l}u’\Big)‘+|A^{\frac{1}{2}}u|^\beta Au+g(u’)=0$inafinitedimensionalHilbertspaceundersuitableassumptionson$g$.