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báo cáo hóa học: " Critical parameter equations for degenerate parabolic equations coupled via nonlinear boundary flux"

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 www.boundaryvalueproblems.eom 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@hotmail.com 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 1.1 coupled via nonlinear boundary flux - up x 0 t uCi 0 t i 1 2.k Uk 1 U1 qk 1 qi 0 t T 1.2 with continuous nonnegative initial data ui x 0 u0i x i 1 2 . k x 0 1.3 compactly supported in R where Pi 1 qi 0 i 1 2 . k are parameters. Parabolic systems like 1.1 - 1.3 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 1.1 - 1.3 that blow up and in the blow-up case the speed at which blowup takes place and the localization of blow-up .