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    Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains

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    Bouzidi_Kheloufi+.pdf (244.0 Kb)
    DOI:
    10.17516/1997-1397-2020-13-3-257-274
    URI (for links/citations):
    http://elib.sfu-kras.ru/handle/2311/135198
    Author:
    Bouzidi, Louanas
    Kheloufi, Arezki
    Бузиди, Луанас
    Хелоуфи, Арезки
    Date:
    2020-05
    Journal Name:
    Журнал Сибирского федерального университета.Математика и физика.Journal of Siberian Federal University. Mathematics & Physics, 2020 13 (3)
    Abstract:
    This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f ∈ L2(D) there exists a unique solution w such that w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [19] in the case of Cauchy-Dirichlet boundary conditions
     
    В этой статье рассматривается параболическое уравнение ∂tw − c(t)∂2xw = f in D, D ={(t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t)}, где φi : [0,+∞[→ R, i = 1, 2 и c : [0,+∞[→ R, удовлетворяя некоторым условиям, задача дополняется граничными условиями типа Дирихле-Робина. Мы изучаем проблему глобальной регулярности в подходящем параболическом пространстве Соболева. В частности, докажем, что для f ∈ L2(D) существует единственное решение w такое, что w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Обратите внимание, что случай ограниченных непрямоугольных областей изучается в [9]. Доказательство основано на оценках энергии после преобразования задачи в полосовой области в сочетании с некоторым интерполяционным неравенством. Эта работа дополняет результаты, полученные в [19] в случае граничных условий Коши-Дирихле
     
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