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1

Source term model for elasticity system with nonlinear dissipative term in a thin domain

Dilmi Mohamed, Dilmi Mourad, Boulaaras Salah, Benseridi Hamid

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

437--451

EN

This article establishes an asymptotic behavior for the elasticity systems with nonlinear source and dissipative terms in a three-dimensional thin domain, which generalizes some previous works. We consider the limit when the thickness tends to zero, and we prove that the limit solution u∗ is a solution of a two-dimensional boundary value problem with lower Tresca’s free-boundary conditions. Moreover, we obtain the weak Reynolds-type equation.

2

The existence of a weak solution of the semilinear first-order differential equation in a Banach space

Jużyniec M.

Czasopismo Techniczne. Nauki Podstawowe

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2014

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R. 111, z. 2-NP

59--62

EN

This paper is devoted to the investigation of the existence and uniqueness of a suitably defined weak solution of the abstract semilinear value problem u'(t) = Au(t) + f(t, u(t)), u(0) = x with x ϵ X, where X is a Banach space. We are concerned with two types of solutions: weak and mild. Under the assumption that A is the generator of a strongly continuous semigroup of linear, bounded operators, we also establish sufficient conditions such that if u is a weak (mild) solution of the initial value problem, then u is a mild (weak) solution of that problem.

PL

Celem pracy jest przedstawienie twierdzenia o jednoznaczności i istnieniu słabego rozwiązania abstrakcyjnego semiliniowego równania różniczkowego u'(t) = Au(t) + f(t, u(t)), u(0) = x, gdzie x ϵ X, w przestrzeni Banacha X. W pracy rozważane są dwa typy rozwiązań: weak oraz mild. Przy założeniu, ze operator A jest generatorem silnie ciągłej półgrupy operatorów liniowych i ograniczonych, podane zostały również warunki wystarczające na to, aby rozwiązanie weak (mild) było rozwiązaniem mild (weak) tego zagadnienia.

3

On differentiability with respect to a parameter of a weak solution of a second order differential equation in Banach space

Jużyniec M.

Czasopismo Techniczne. Nauki Podstawowe

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2010

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R. 107, z. 1-NP

31-41

EN

The purpose of this paper is to present some theorems on continuity and differentiability with respect to a parameter h of a weak solution of the evolution equation u(t) = A[h]u(t) + f[h](t).

PL

Celem artykułu jest przedstawienie twierdzeń o ciągłej zależności od parametru oraz różniczkowalności względem parametru h słabego rozwiązania ewolucyjnego równania różniczkowego u(t) = A[h]u(t) + f[h](t).

4

A quasistatic contact problem with slip dependent coefficient of friction for elastic materials

Corneschi C., Hoarau-Mantel T.-V. T.-V., Sofonea M.

Journal of Applied Analysis

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2002

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Vol. 8, nr 1

63-82

EN

We consider a mathematical model which describes the frictional contact between a deformable body and an obstacle, say a foundation. The body is assumed to be linear elastic and the contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. The novelty consists here in the fact that we consider a slip dependent coefficient of friction and a quasistatic process. We present two alternative yet equivalent formulations of the problem and establish existence and uniqueness results. The proofs are based on a new result obtained in [10] in the study of evolutionary variational inequalities.

5

Chemical attack in free boundary domains

Clain S.

Journal of Applied Analysis

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1999

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Vol. 5, nr 1

35-58

EN

This paper presents a mathematical model for a chemical process used to machine cristal as glass or silica. A short physical description is presented from which we draw the mathematical model. We obtain a coupled parabolic equations system on a free boundary domain with a non-linear condition on the boundary. The existence and the uniqueness is proved in the one-dimensional case.