Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Critical parameter equations for degenerate parabolic equations coupled via nonlinear boundary flux | Xu and Song Boundary Value Problems 2011 2011 15 http content 2011 1 15 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Critical parameter equations for degenerate parabolic equations coupled via nonlinear boundary flux Si Xu and Zifen Song Correspondence xusi_math@ Department of Mathematics Jiangxi Vocational College of Finance and Economics Jiujiang Jiangxi 332000 PR China Abstract This paper deals with the critical parameter equations for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution we obtain the critical global existence parameter equation. The critical Fujita type is conjectured with the aid of some new results. Mathematics Subject Classification 2000 . 35K55 35K57. Keywords degenerate parabolic system global existence blow-up 1 Introduction In this paper we consider the following degenerate parabolic equations dt upi xx i 1 2 . k x 0 0 t T coupled via nonlinear boundary flux - up x 0 t uCi 0 t i 1 Uk 1 U1 qk 1 qi 0 t T with continuous nonnegative initial data ui x 0 u0i x i 1 2 . k x 0 compactly supported in R where Pi 1 qi 0 i 1 2 . k are parameters. Parabolic systems like - appear in several branches of applied mathematics. They have been used to models for example chemical reactions heat transfer or population dynamics see 1 and the references therein . As we shall see under certain conditions the solutions of this problem can become unbounded in a finite time. This phenomenon is known as blow-up and has been observed for several scalar equations since the pioneering work of Fujita 2 . For further references see the review by Leivine 3 . Blow-up may also happen for systems see 4-7 . Our main interest here will be to determine under which conditions there are solutions of - that blow up and in the blow-up case the speed at which blowup takes place and the localization of blow-up .