Long-time behavior of reaction–diffusion equations with dynamical boundary condition

2011 ◽  
Vol 74 (12) ◽  
pp. 3876-3883 ◽  
Author(s):  
Lu Yang ◽  
Meihua Yang
Keyword(s):  
Boundary Condition ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamical Boundary Condition ◽  
Long Time

Long-time behavior of solutions and chaos in reaction-diffusion equations

2017 ◽  
Vol 99 ◽  
pp. 91-100 ◽  
Author(s):  
Kamal N. Soltanov ◽  
Anatolij K. Prykarpatski ◽  
Denis Blackmore
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time

LONG-TIME BEHAVIOR OF SOLUTIONS TO NONLINEAR REACTION DIFFUSION EQUATIONS INVOLVING L1 DATA

2012 ◽  
Vol 14 (01) ◽  
pp. 1250007 ◽  
Author(s):  
CHENGKUI ZHONG ◽  
WEISHENG NIU
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Smooth Bounded Domain ◽  
Bootstrap Argument ◽  
Nonlinear Reaction ◽  
L1 Data ◽  
Long Time

In this paper we consider the long-time behavior of solutions to nonlinear reaction diffusion equations involving L1 data, [Formula: see text] where Ω is a smooth bounded domain and u0, g ∈ L1(Ω). Using a decomposition technique combined with a bootstrap argument we establish some uniform regularity results on the solutions, by which we prove that the solution semigroup generated by the problem above possesses a global attractor [Formula: see text] in L1(Ω). Moreover, we obtain that the attractor is actually invariant, compact in [Formula: see text], q < max {N/(N-1), (2p-2)/p}, and attracts every bounded subset of L1(Ω) in the norm of [Formula: see text], 1 ≤ r < ∞.

scholarly journals On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Anh Cung The ◽  
Toi Vu Manh
Keyword(s):  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Reaction Diffusion Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Semilinear Parabolic Systems ◽  
Long Time ◽  
Grushin Operators ◽  
Finite Fractal Dimension ◽  
Semilinear Parabolic

We study the long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullbackD-attractor in(L2(Ω))mfor the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullbackD-attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.

scholarly journals Asymptotic Behavior of Solutions to Reaction-Diffusion Equations with Dynamic Boundary Conditions and Irregular Data

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Yonghong Duan ◽  
Chunlei Hu ◽  
Xiaojuan Chai
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Large Time Behavior ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Dynamic Boundary

This paper is concerned with the asymptotic behavior of solutions to reaction-diffusion equations with dynamic boundary conditions as well as L1-initial data and forcing terms. We first prove the existence and uniqueness of an entropy solution by smoothing approximations. Then we consider the large-time behavior of the solution. The existence of a global attractor for the solution semigroup is obtained in L1(Ω¯,dν). This extends the corresponding results in the literatures.

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