We study the positive blowing-up solutions of the semilinear parabolic system: , where and . We prove that if or then one component of a blowing-up solution may stay bounded until the blow-up time, while if and this cannot happen. We also investigate the blow up rates of a class of positive radial solutions. We prove that in some range of the parameters and , solutions of the system have an uncoupled blow-up asymptotic behavior, while in another range they have a coupled blow-up behavior.
"Optimal condition for non-simultaneous blow-up in a reaction-diffusion system." J. Math. Soc. Japan 56 (2) 571 - 584, April, 2004. https://doi.org/10.2969/jmsj/1191418646