Weighted Gagliardo–Nirenberg Inequalities Involving BMO Norms and Solvability of Strongly Coupled Parabolic Systems

2016 ◽  
Vol 16 (1) ◽  
pp. 125-146 ◽  
Author(s):  
Dung Le
Keyword(s):  
Global Existence ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Parabolic Systems ◽  
Strongly Coupled ◽  
Global Existence Of Solutions ◽  
Coupled Parabolic Systems

AbstractNew weighted Gagliardo–Nirenberg inequalities are introduced together with applications to the local/global existence of solutions to nonlinear strongly coupled and uniform parabolic systems. Much weaker sufficient conditions than those existing in literature for solvability of these systems will be established.

Global Existence Results for Near Triangular Nonlinear Parabolic Systems

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Dung Le
Keyword(s):  
Global Existence ◽  
Weak Solutions ◽  
Parabolic Systems ◽  
Existence Results ◽  
Nonlinear Parabolic Systems ◽  
Strongly Coupled ◽  
Nonlinear Parabolic ◽  
Regularity Of Weak Solutions ◽  
Coupled Parabolic Systems

AbstractWe study the global existence and regularity of weak solutions to strongly coupled parabolic systems whose diffusion matrices are almost triangular.

Über Differenzenverfahren zur numerischen Lösung stark gekoppelter Systeme nichtlinearer Diffusionsgleichungen

1987 ◽  
Vol 42 (10) ◽  
pp. 1133-1140 ◽  
Author(s):  
Karl Graf Finck von Finckenstein
Keyword(s):  
Nonlinear System ◽  
Main Part ◽  
Parabolic Systems ◽  
Difference Operators ◽  
Discrete Approximations ◽  
Strongly Coupled ◽  
One Step ◽  
Difference Methods ◽  
Coupled Parabolic Systems

A class of nonlinear implicit one step difference methods for quasilinear strongly coupled parabolic systems in two space variables is considered. The main part of the paper deals with proving convergence of the discrete approximations for vanishing step sizes. For this purpose, bounds for the inverse difference operators have to be derived previously. This is possible subject to a condition which can be considered as a generalization of the concept “parabolic” to systems. Finally, it is shown that the nonlinear system s arising from the discretizations have one and only one solution for all step sizes being sufficiently small.

Quantitative maximum principles and strongly coupled gradient-like reaction-diffusion systems

1983 ◽  
Vol 94 (3-4) ◽  
pp. 265-286 ◽  
Author(s):  
Nicholas D. Alikakos
Keyword(s):  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Diffusion Matrix ◽  
Strongly Coupled ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
All Solutions

SynopsisIn §§1 and 2, we consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at t →∞. The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. In §3, we establish uniform L§ bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.

Counterexamples for uniqueness in strongly coupled parabolic systems

1980 ◽  
Vol 171 (1) ◽  
pp. 83-90
Author(s):  
Ray Redheffer ◽  
Wolfgang Walter
Keyword(s):  
Parabolic Systems ◽  
Strongly Coupled ◽  
Coupled Parabolic Systems
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