scholarly journals Sturm global attractors for $S^1$-equivariant parabolic equations

2012 ◽  
Vol 7 (4) ◽  
pp. 617-659 ◽  
Author(s):  
Bernold Fiedler ◽  
◽  
Carlos Rocha ◽  
Matthias Wolfrum ◽  
Keyword(s):  
Parabolic Equations ◽  
Global Attractors

scholarly journals Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yanjun Zhang ◽  
Qiaozhen Ma
Keyword(s):  
Parabolic Equations ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Regularity Result ◽  
Global Attractors ◽  
Reaction Diffusion Equation ◽  
Exponential Attractors ◽  
Finite Dimensional ◽  
Fixed Subset ◽  
Two Parameters

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations ut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.

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