Coupled Versus Uncoupled Blow-Up Rates in Cooperative n-Species Logistic Systems

Abstract

This paper ascertains the exact boundary blow-up rates of the large positive solutions of a class of cooperative logistic systems involving n species in a general domain of ℝN{\mathbb{R}^{N}} of class 𝒞2+ν{\mathcal{C}^{2+\nu}}, 0<ν<1{0<\nu<1}. The problem models a population divided in groups whose individuals compete with those of the same group, while simultaneously they cooperate with the members of the remaining groups. Our main result provides with the exact blow-up rates along the edges of Ω from the values of the blow-up rates of the underlying uncoupled system. Rather astonishingly, these blow-up rates are independent of the strength of the cooperative effects, which play a secondary role in the analysis carried out in this paper. No previous result of this nature is available in the specialized literature for more than n=2{n=2} species.

Keywords

• cooperative systems
• large solutions
• blow-up rates on the boundary
• vanishing weights at variable rates
• uniqueness of large solutions
• 35j57
• 35k57
• 35b44
• 35q92