%0 Journal Article
%A Arrieta Algarra, José María
%A Carvalho, Alexandre N.
%A Rodríguez Bernal, Aníbal
%T Attractors of parabolic problems with nonlinear boundary conditions uniform bounds
%D 2000
%@ 0360-5302
%U https://hdl.handle.net/20.500.14352/57904
%X 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| .
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