Attractors of parabolic problems with nonlinear boundary conditions uniform bounds

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| .