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Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary

José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira

2017
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Discrete and continuous dynamical systems. Series B
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been set as a strip without oscillatory behavior in a fixed domain Ω. Later, the dynamical system given by a semilinear parabolic problem in the same situation was analyzed in [20, 21] where the upper semicontinuity of attractors at ε = 0 has been shown. In [3, 4] the results of [11, 20] were extended to a reaction-diffusion problem with delay. In these works, the boundary of the domain is always assumed to be smooth. Subsequently some results of [11] were adapted in [5] to be considered in a
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... milinear elliptic problem posed on a Lipschitz fixed domain Ω with the ε-neighborhood presenting highly oscillatory behavior. The upper and lower semicontinuity of the attractor to the associated parabolic problem in smooth fixed domains were shown in [6] . Recently, some results from [11, 5] have been adapted in [1, 2] to a class of narrow strips θ ε and bounded oscillatory domains Ω ε . Under the restricted assumption Ω ⊂ Ω ε and θ ε ⊂ Ω ε \ Ω for all ε > 0, the authors have been able to estimate concentrating integrals and analyze the asymptotic behavior of semilinear elliptic equations as Ω ε → Ω and ∂Ω ε → ∂Ω when ε → 0 in the sense of Hausdorff. This paper is organized as follows: in Section 2 we introduce the assumptions, notations and the main result. In Section 3, we show some technical results concerning extension operators, Lebesgue-Bochner and Sobolev-Bochner generalized spaces needed to get our estimates. Following by Section 4, we prove some properties about concentrating integrals which are used in Section 5 to study the nonlinearities of our problem. Finally, in Section 6, we pass to the limit in a semilinear elliptic problem getting the upper semicontinuity of the solutions. Moreover, assuming hyperbolicity to the solutions of the limit equation, we also obtain the lower semicontinuity at ε = 0, and we will exclude the possibility that, near an equilibrium point of the limiting equation, may exist several different equilibrium points of the perturbed problem, and therefore, we will also prove some sort of uniqueness of the equilibrium points. 2. Assumptions, notations and main result. To fix the problem, notation and main hypotheses, let us start considering problem (1) where

doi:10.3934/dcdsb.2019079
fatcat:uhdsjqvtdngijf3b7em4qpr6zq