scholarly journals Periodic solutions of a class of degenerate parabolic system with delays

2011 ◽  
Vol 380 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Yifu Wang ◽  
Jingxue Yin
Keyword(s):  
Periodic Solutions ◽  
Parabolic System ◽  
Degenerate Parabolic System ◽  
Degenerate Parabolic

scholarly journals Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yan Dong ◽  
Guangwei Du ◽  
Kelei Zhang
Keyword(s):  
Parabolic System ◽  
Weak Solutions ◽  
Vector Fields ◽  
Degenerate Parabolic System ◽  
Higher Integrability ◽  
Singular Coefficients ◽  
Degenerate Parabolic ◽  
Alpha Beta ◽  
Hörmander’S Condition ◽  
Reverse Hölder

Abstract In this paper, we study the degenerate parabolic system $$ u_{t}^{i} + X_{\alpha }^{*} \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$ u t i + X α ∗ ( a i j α β ( z ) X β u j ) = g i ( z , u , X u ) + X α ∗ f i α ( z , u , X u ) , where $X=\{X_{1},\ldots,X_{m} \}$ X = { X 1 , … , X m } is a system of smooth real vector fields satisfying Hörmander’s condition and the coefficients $a_{ij}^{\alpha \beta }$ a i j α β are measurable functions and their skew-symmetric part can be unbounded. After proving the $L^{2}$ L 2 estimates for the weak solutions, the higher integrability is proved by establishing a reverse Hölder inequality for weak solutions.

scholarly journals Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Ling Zhengqiu ◽  
Wang Zejia
Keyword(s):  
Global Existence ◽  
Critical Exponent ◽  
Parabolic System ◽  
Blow Up ◽  
Nonnegative Solution ◽  
Degenerate Parabolic System ◽  
Nonlocal Sources ◽  
Nonnegative Solutions ◽  
Large Ball ◽  
Degenerate Parabolic

This paper investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, ifPc=p1q1−(m−p2)(n−q2)<0, then every nonnegative solution is global, whereas ifPc>0, there are solutions that blowup and others that are global according to the size of initial valuesu0(x)andv0(x). WhenPc=0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

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