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Nonlocal evolution equations in perforated domains
Author(s) -
Pereira Marcone C.
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5144
Subject(s) - mathematics , limit (mathematics) , invertible matrix , evolution equation , kernel (algebra) , domain (mathematical analysis) , convergence (economics) , mathematical analysis , function (biology) , pure mathematics , evolutionary biology , economics , biology , economic growth
In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form u t ( t , x ) = ∫ R N ∖ A ϵJ ( x − y ) ( u ( t , y ) − u ( t , x ) ) d y + f ( t , x ) with x in a perturbed domain Ω ϵ ⊂ Ω . We think about Ω ε as a fixed set Ω from where we have removed the subset A ε that we call the holes. Moreover, we take J as a nonsingular kernel. Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω ε has a weak limit, χ ϵ ⇀ X weakly ∗ in L ∞ (Ω) as ε →0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation.