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Владикавказский математический журнал

Vladikavkaz Mathematical Journal

Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН)

Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Absorption and Nonlinear Nonlocal Boundary Conditions

Начально-краевая задача для системы полулинейных параболических уравнений с поглощением и нелинейными нелокальными граничными условиями

10.46698/o2548-3794-2309-a

16691

06 2025

27 2 5 18

https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=16755&SECTION_ID=631

Булыно

Д. А.

Bulyno

D. A.

dabulyno13@gmail.com

Гладков

А. Л.

Gladkov

A. L.

gladkoval@bsu.by

Никитин

А. И.

Nikitin

A. I.

ip.alexnikitin@gmail.com

Белорусский государственный университет, Беларусь, 220030, Минск, пр. Независимости, 4 Belarusian State University, 4 Nezavisimosti Ave., Minsk 220030, Belarus

Витебский государственный университет им. П. М. Машерова, Беларусь, 210038, Витебск, Московский проспект, 33 Masherov Vitebsk State University, 33 Moskovskiy Ave., Vitebsk 210038, Belarus

In this paper we consider classical solutions of an initial boundary value problem for a system of semilinear parabolic equations with absorption and nonlinear nonlocal boundary conditions. Nonlinearities in equations and boundary conditions may not satisfy the Lipschitz condition. To prove the existence of a solution we regularize the original problem. Using the Schauder-Tikhonov fixed point theorem, the existence of a local solution of regularized problem is proved. It is shown that the limit of solutions of the regularized problem is a maximal solution of the original problem. Using the properties of a maximal solution, a comparison principle is proved. In this case, no additional assumptions are made when nonlinearities in absorption do not satisfy the Lipschitz condition. Conditions are found under which solutions are positive functions. The uniqueness of the solution is established. It is shown that the trivial solution \((0, 0)\) may not be unique.

В работе мы рассматриваем классические решения начально-краевой задачи для системы полулинейных параболических уравнений с поглощением и нелинейными нелокальными граничными условиями. Нелинейности в уравнениях и граничных условиях могут не удовлетворять условию Липшица. Для доказательства существования решения мы регуляризуем исходную задачу. Используя теорему Шаудера - Тихонова о неподвижной точке, доказывается существование локального решения регуляризованной задачи. Показано, что предел решений регуляризованной задачи является максимальным решением исходной задачи. Используя свойства максимального решения, доказывается принцип сравнения. При этом не делается дополнительных предположений, когда нелинейности в поглощении не удовлетворяют условию Липшица. Найдены условия, при выполнении которых решения являются положительными функциями. Устанавливается единственность решения. Показано, что нулевое решение может быть неединственным.

система полулинейных параболических уравнений нелокальные граничные условия существование решения принцип сравнения

system of semilinear parabolic equations nonlocal boundary conditions existence of a solution comparison principle

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