Critical Curves For Degenerate Parabolic Equations Coupled By Boundary Sources

Nonlinear diffusion equations are derived from widespread diffusion phenomenonin nature, such as filtration, image processing, population ecology and dynamics ofbiological groups, and so on. In this paper, we investigate the large time behaviorof solutions to nonlinear diffusion equations, including global existence in time andblow-up in a finite time. In particular, we are interested in the critical global existencecurve and critical Fujita curve which can be used to describe the large time behavior ofsolutions exactly.In recent years, related studies have shown that, the critical global existence expo-nent and the critical Fujita exponent are not equal for one dimensional case.However,under some conditions, the critical global existence exponent and the critical Fujita ex-ponent are equal for high dimensional case. In the corresponding by boundary sourcecoupled diffusion equations in the study,there exist the critical global existence curveand the critical Fujita curve, which describe asymptotic behavior of the solutions,suchas global existence in time and blow-up in a finite time. In [1],the authors studied new-tonian filtration equations in high dimensional case and proved that the critical glob-al existence curve coincides with the critical Fujita curve under some conditions. Inthis paper,the results are generalized to non-newtonian filtration equations and non-newtonian polytropic filtration equations.We discuss global existence in time and blow-up in a finite time of the solutions, mainly by means of upper and lower solutions approaeh and comparison principle.Finally,weprovethatthecriticalglobalexistencecurvecoincideswiththecriticalFujitacurve under some conditions for non-newtonian filtration equations and non-newtonianpolytropic filtration equations in high dimensional case.The paper is divided into four chapters. In the first chapter,we recall the back-groundandthecurrentdevelopmentoftherelatedtopic,andintroducesomepreliminaryknowledge. In the second chapter,we study critical curves for high dimensional fast d-iffusion non-newtonian filtration equations coupled by boundary sources. In the third chapter,we consider critical curves for high dimensional fast diffusion non-newtonianpolytropic filtration equations coupled by boundary sources. In the fourth chapter,wefocus on critical curves for high dimensional fast diffusion non-newtonian polytropicfiltration equations coupled by boundary sources with parameters.where1<p,q <2, α,β≥0, N≥2, B1(0) is the unit ball in RNwith boundary B1(0), ν is the inward normal vector on B1(0), and u0(x), v0(x) are nonnegative, suitablysmooth and bounded functions satisfying the appropriate compatibility conditions. Weget that the critical global existence curve and the critical Fujita curve for the abovingproblem arewhere m,n>0, p,q>1,0<m(p1),n(q1)<1, α,β>0, N≥2. B1(0) isthe unit ball in RNwith boundary B1(0), ν is the inward normal vector on B1(0), and u0(x), v0(x) are nonnegative, suitably smooth and bounded functions satisfying theappropriate compatibility conditions. We get that if N> max{p,q},the critical globalexistence curve and the critical Fujita curve for the aboving problem arewhere m,n>0, p,q>1,0<m(p1),n(q1)<1, α,β>0,α1, β1≥0, N≥2. B1(0)is the unit ball in RNwith boundary B1(0), ν is the inward normal vector on B1(0),and u0(x), v0(x) are nonnegative, suitably smooth and bounded functions satisfying theappropriate compatibility conditions. We get that if N> max{p,q},α1≤m(p1),β1≤n(q1),the critical global existence curve and the critical Fujita curve for the abovingproblem are...