In this paper we analyze the behavior of the Laplace operator with Neumann
boundary conditions in a thin domain of the type $R^\epsilon = \{(x,y) \in
\R^2; x \in (0,1), 0 < y < \epsilon G(x, x/\epsilon)\} $ where the function
G(x,y) is periodic in y of period L. Observe that the upper boundary of the
thin domain presents a highly oscillatory behavior and, moreover, the height of
the thin domain, the amplitude and period of the oscillations are all of the
same order, given by the small parameter $\epsilon$.Comment: 27 pages, 2 figure
In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type Rwhere the function G is supposed to be l(x)-periodic in the second variable y. The condition α > 1 implies that the upper boundary of this thin domain presents a very high oscillatory behavior. Indeed, we have that the order of its oscillations is larger than the order of the amplitude and height of R given by the small parameter . We also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions, but rather more comb-like domains.
In this work we show, for a class of dissipative semilinear parabolic problems, that the global compact attractor varies continuously with respect to parameters in the equations. Applications to a parabolic problem with nonlinear boundary conditions are also obtained.
Abstract. In this paper we investigate the behavior of a family of steady state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a -neighborhood of a portion Γ of the boundary. We assume that this -neighborhood shrinks to Γ as the small parameter goes to zero. Also, we suppose the upper boundary of this -strip presents a highly oscillatory behavior. Our main goal here is to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood.
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