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Attractors of parabolic problems with nonlinear boundary conditions uniform bounds

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Arrieta Algarra, José María and Carvalho, Alexandre N. and Rodríguez Bernal, Aníbal (2000) Attractors of parabolic problems with nonlinear boundary conditions uniform bounds. Communications in Partial Differential Equations, 25 (1-2). pp. 1-37. ISSN 0360-5302

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Official URL: http://www.tandfonline.com/doi/abs/10.1080/03605300008821506


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Abstract

The authors study the asymptotic behavior of solutions to a semilinear parabolic problem u t −div(a(x)∇u)+c(x)u=f(x,u) for u=u(x,t), t>0, x∈Ω⊂⊂R N , a(x)>m>0; u(x,0)=u 0 with nonlinear boundary conditions of the form u=0 on Γ 0 , and a(x)∂ n u+b(x)u=g(x,u) on Γ 1 , where Γ i are components of ∂Ω . Under smoothness and growth conditions which ensure the local classical well-posedness of the problem, they indicate some sign conditions under which the solutions are globally defined in time, and somewhat more strong dissipativeness conditions under which they possess a global attractor that captures the asymptotic dynamics of the system. After that the authors study the dependence of the attractors on the diffusion. For a(x)=a ε (x) they show their upper semicontinuity on ε . Throughout the paper they also pay special attention to the dependence of the estimates obtained on the domain Ω and show that in certain instances the L ∞ bounds on the attractors do not depend on the shape of Ω but rather on |Ω| .


Item Type:Article
Uncontrolled Keywords:Semilinear equation; Groth restrictions; Sign conditions; Dissipativeness condition
Subjects:Sciences > Mathematics > Differential equations
ID Code:19910
Deposited On:12 Feb 2013 16:30
Last Modified:12 Dec 2018 15:07

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