Some results on the existence of weak periodic solutions for quasilinear parabolic systems with L1 data

The aim of this paper is to prove the existence of weak periodic solution and super solution for M×M reaction diffusion system with L1 data and nonlinearity on the gradient. The existence is proved by the technique of sub and super solution and Schauder fixed point theorem.

AN EXISTENCE RESULT FOR QUASILINEAR PARABOLIC SYSTEMS WITH LOWER ORDER TERMS

In this paper we prove the existence of weak solutions for a class of quasilinear parabolic systems, which correspond to diffusion problems, in the form where Ω is a bounded open domain of be given and The function v belongs to is in a moving and dissolving substance, the dissolution is described by f and the motion by g. We prove the existence result by using Galerkin’s approximation and the theory of Young measures.

Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type

In this paper, the Hölder regularity of weak solutions for singular parabolic systems of p-Laplacian type is investigated. By the Poincare inequality, we show that its weak solutions within Hölder space.

On boundary optimal control of a coefficient in a nonlinear parabolic equation

Работа связана с изучением нелинейных параболических систем, возникающих при моделировании и управлении физико-химическими процессами, в которых происходят изменения внутренних свойств материалов. Исследовано оптимальное управление одной из таких систем, которая включает в себя краевую задачу третьего рода для квазилинейного параболического уравнения с неизвестным коэффициентом при производной по времени, а также уравнение изменения по времени этого коэффициента. Обоснована постановка оптимальной задачи с финальным наблюдением искомого коэффициента, в которой управлением является граничный режим на одной из границ области. Получено явное представление дифференциала минимизируемого функционала через решение сопряженной задачи. Доказаны условия ее однозначной разрешимости в классе гладких функций. Полученные результаты имеют практическое значение для приложений в различных технических областях, медицине, геологии и т.п. Приведены некоторые примеры таких приложений. The work is connected with investigation of nonlinear parabolic systems arising in the mathematical modeling and control of physical-chemical processes in which inner properties of materials are subjected to changes. We consider optimal control in one of such systems that involves a boundary value problem of the third kind for a quasilinear parabolic equation with an unknown coefficient at the time derivative and, moreover, an additional equation for a time dependence of this coefficient. The optimal problem with a boundary control regime is justified for the given final observation of the sought coefficient. The exact representation for the differential of the minimization functional in terms of the solutions of the conjugate problem is obtained. The form of this conjugate problem and conditions of unique solvability in a class of smooth functions are shown. The obtained results are important for applications in various technical fields, medicine, geology, etc. Some examples of such applications are discussed.