Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type

2015 ◽  
Vol 147 (3-4) ◽  
pp. 311-363 ◽  
Author(s):  
Yoshie Sugiyama
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Blow Up ◽  
Parabolic Systems ◽  
Degenerate Parabolic Systems ◽  
Degenerate Parabolic ◽  
Medium Type

scholarly journals Lp Estimates for Weak Solutions to Nonlinear Degenerate Parabolic Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Na Wei ◽  
Xiangyu Ge ◽  
Yonghong Wu ◽  
Leina Zhao
Keyword(s):  
Weak Solutions ◽  
Vector Fields ◽  
Parabolic Systems ◽  
Growth Conditions ◽  
Lp Estimates ◽  
Reverse Hölder Inequalities ◽  
Degenerate Parabolic Systems ◽  
Degenerate Parabolic ◽  
Reverse Hölder ◽  
Nonlinear Degenerate

This paper is devoted to the Lp estimates for weak solutions to nonlinear degenerate parabolic systems related to Hörmander’s vector fields. The reverse Hölder inequalities for degenerate parabolic system under the controllable growth conditions and natural growth conditions are established, respectively, and an important multiplicative inequality is proved; finally, we obtain the Lp estimates for the weak solutions by combining the results of Gianazza and the Caccioppoli inequality.

scholarly journals Hölder regularity for porous medium systems

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Naian Liao
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Parabolic Systems ◽  
Type Result ◽  
Local Boundedness ◽  
Liouville Type ◽  
Medium Type ◽  
Hölder Estimates

AbstractWe establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. $$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$ ∂ t u - div ( m | u | m - 1 D u ) = 0 , m > 0 . As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

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