Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains
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https://elib.sfu-kras.ru/handle/2311/135198Author:
Bouzidi, Louanas
Kheloufi, Arezki
Бузиди, Луанас
Хелоуфи, Арезки
Date:
2020-05Journal 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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