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: Non-Newtonian polytropic filtration systems with nonlinear boundary conditions | Du and Li Boundary Value Problems 2011 2011 2 http content 2011 1 2 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Non-Newtonian polytropic filtration systems with nonlinear boundary conditions Wanjuan Du and Zhongping Li Correspondence duwanjuan28@ College of Mathematic and Information China West Normal University Nanchong 637002 PR China SpringerOpen0 Abstract This article deals with the global existence and the blow-up of non-Newtonian polytropic filtration systems with nonlinear boundary conditions. Necessary and sufficient conditions on the global existence of all positive weak solutions are obtained by constructing various upper and lower solutions. Mathematics Subject Classification 2000 35K50 35K55 35k6s Keywords Polytropic filtration systems Nonlinear boundary conditions Global existence Blow-up Introduction In this article we study the global existence and the blow-up of non-Newtonian polytropic filtration systems with nonlinear boundary conditions uk t AmiUi i 1 . n x e Q t 0 Vmiui V n j i 1 . n x e do t 0 1 ui x 0 ui0 x 0 i 1 . n x e Í2 where N Amui div V ui m -1V ui V V ui m -1uix Vmui V ui m -1uix . V ui m -1uix m i i m i m i ux I ux Q c Rn is a bounded domain with smooth boundary 3Q V is the outward normal vector on the boundary dQ and the constants ki mi 0 m j 0 i j 1 . n ui0 x i 1 . n are positive C1 functions satisfying the compatibility conditions. The particular feature of the equations in is their power- and gradient-dependent diffusibility. Such equations arise in some physical models such as population dynamics chemical reactions heat transfer and so on. In particular equations in may be used to describe the nonstationary flows in a porous medium of fluids with a power dependence of the tangential stress on the velocity of displacement under polytropic conditions. In this case the equations in are called the non-Newtonian polytropic filtration equations which have .
