On p-homogeneous systems of differential equations and their linear perturbations

Abstract

Long-time asymptotic properties of the solutions of the system

where f (u) is positive homogeneous of degree p>1, are studied. We also consider the corresponding linearly perturbed system

It is shown that if A=α I, then the global existence of all solutions for one value of α implies that the same property holds for all α, and that all solutions converge to the origin when α<0. On the other hand, it is shown that the addition of a matrix for which A u is inward in the sense that u · A u <0 for u ≠ 0 can turn a p-homogeneous system all of whose solutions are bounded into one which has solutions which blow up in a finite time.

Keywords:

• Growup
• Blowup
• Linear perturbation
• Inward perturbation

Keywords:

• 2000 Mathematics Subject Classifications: 34A34
• 34C11