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1

On the solvability of some parabolic equations involving nonlinear boundary conditions with L1 data

Taourirte Laila, Charkaoui Abderrahim, Alaa Nour Eddine

Opuscula Mathematica

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2024

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Vol. 44, no. 4

587--623

EN

We analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and L1 data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder’s topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models.

2

Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Godoy Tomas

Opuscula Mathematica

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2023

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Vol. 43, no. 1

19--46

EN

Let Ω be a C2 bounded domain in Rn such that ∂Ω = Γ1 ∪ Γ2, where Γ1 and Γ2 are disjoint closed subsets of ∂Ω, and consider the problem −Δu = g(·, u) in Ω, u = τ on Γ1, ∂u ∂ν = η on Γ2, where 0 ≤ τ ∈ W 1 2 ,2(Γ1), η ∈ (H1 0, Γ1(Ω))′, and g : Ω×(0,∞) → R is a nonnegative Carathéodory function. Under suitable assumptions on g and η we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow g to be singular at s = 0 and also at x ∈ S for some suitable subsets S ⊂ Ω. The Dirichlet problem −Δu = g(·, u) in Ω, u = σ on ∂Ω is also studied in the case when 0 ≤ σ ∈ W 1 2 ,2(Ω).

3

Exponential decay of solutions to a class of fourth-order nonlinear hyperbolic equations modeling the oscillations of suspension bridges

Liu Yang, Yang Chao

Opuscula Mathematica

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2022

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Vol. 42, no. 2

239--255

EN

This paper is concerned with a class of fourth-order nonlinear hyperbolic equations subject to free boundary conditions that can be used to describe the nonlinear dynamics of suspension bridges.

4

A note on the regularity of weak solutions to the coupled 2D Allen-Cahn-Navier-Stokes system

Medjo T. Tachim

Journal of Applied Analysis

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2019

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Vol. 25, nr 1

111--117

EN

In this article,we study a coupled Allen-Cahn-Navier-Stokes model in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter.We present an equivalent weak formulation for the model, and we prove a new regularity result for the weak solutions.

5

Asymptotic of weak solutions for linear elliptic nonlocal Robin problem without Dini-continuity condition in a plane angle domain

Żyjewski K.

Commentationes Mathematicae

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2017

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Vol 57, No. 1

9--24

EN

We investigate the behaviour of weak solutions to the nonlocal Robin problem for linear elliptic divergence second order equations in a neighbourhood of the boundary corner point. We find the exponent of the solution decreasing rate under the assumption that the leading coefficients of the equations do not satisfy the Dini-continuity condition.

6

Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

Kong L.

Opuscula Mathematica

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2016

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Vol. 36, no. 2

253--264

EN

We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a p(x)-biharmonic operator [formula] where Ω is a smooth bounded domain in RN, [formula] is the p(x)-biharmonic operator, and λ > 0 is a parameter. We establish sufficient conditions under which there exists a positive number λ* such that the above problem has at least two nontrivial weak solutions for each λ > λ*. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces [formula].

7

Existence and regularity of solutions for hyperbolic functional differential problems

Kamont Z.

Opuscula Mathematica

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2014

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Vol. 34, no. 2

217--242

EN

A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.

8

A Neumann boundary value problem for a class of gradient systems

Pan W. W., Li L.

Opuscula Mathematica

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2014

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Vol. 34, no. 1

171--181

EN

In this paper we study a class of two-point boundary value systems. Using very recent critical points theorems, we establish the existence of one non-trivial solution and infinitely many solutions of this problem, respectively.

9

On the solvability of Dirichlet problem for the weighted p-Laplacian

Szlachtowska E.

Opuscula Mathematica

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2012

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Vol. 32, no. 4

775-781

EN

The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted ρ-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.

10

A coupled system of fractional order integral equations in reflexive Banach spaces

El-Sayed A.M.A., Hashem H.H.G.

Commentationes Mathematicae

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2012

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Vol. 52, [Z] 1

21-28

EN

We present an existence theorem for at least one weak solution for a coupled system of integral equations of fractional order in reflexive Banach spaces relative to the weak topology.

11

Pewne zastosowania w elektrotermii twierdzenia Banacha o punkcie stałym oraz słabych rozwiązań równań różniczkowych cząstkowych

Brodzki M.

Prace Naukowe Politechniki Śląskiej. Elektryka

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2008

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z. 2

7-32

PL

Praca ta poświęcona jest zastosowaniu twierdzenia Banacha o punkcie stałym oraz pojęcia rozwiązań słabych do najprostszych problemów elektrotermii. Dla ośrodków elektrycznie i termicznie liniowych jednorodnych i izotropowych spoczywających w inercjalnym układzie odniesienia, sformułowane są w obszarze przestrzennym o gładkim brzegu i dla czasów z ograniczonego przedziału, przy użyciu kartezjańskich układów współrzędnych, równania potencjału wektorowego pola magnetycznego oraz równania przewodzenia ciepła ze standardowymi warunkami granicznymi dla tego typu równań. Równanie potencjału przy zadanej funkcji temperatury jest liniowe o zmiennych współczynnikach. Równanie termiczne przy zadanym potencjale wektorowym pola magnetycznego jest już samo nieliniowe. Sformułowane jest pojęcie słabych rozwiązań dla takich równań. Rozwiązania te (problemów elektrycznych i termicznych z osobna) wyznaczane są w przybliżeniu z zastosowaniem metody Galerkina oraz twierdzenia o punkcie stałym. Następnie konstruowane jest odwzorowanie zbliżające dla wspólnego problemu elektrotermicznego. Wyznaczone są stałe zbliżania i czas jego obserwacji oraz objaśnione zastosowanie twierdzenia Banacha. Następnie przedyskutowane są otrzymane rezultaty oraz perspektywy zastosowania układów współrzędnych przestrzennie krzywoliniowych.

EN

This work deals with some applications of Banach theorem of fixed point and weak solutions in simple problems of electrothermics. For electrically and thermally linear homogeneous and isotropic medium remaining in inertial reference system equations of magnetic field vector potential and a heat equation with standard boundary conditions can be formulated for a range with smooth edge and for limited time period. Potential equation for a given temperature function is linear with variable coefficients while heat equation for a given vector potential of magnetic field is nonlinear. For such equations weak solutions are formulated. Their approximations are separately found for electric and thermal problems using Galerkin method and fixed point theorem. Then approaching mapping is constructed for a joint electrothermal problem. After that approach constants and observation time are determined followed by some explanation on Banach theorem application. Finally, obtained results as well as future application of curvilinear coordination systems are discussed

12

On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

Gomaa A,M

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2007

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Vol. 47, [Z] 2

179-191

EN

In the present work we give an existence theorem for bounded weak solution of the differential equation.......[formuła matematyczna]

13

On the nonlinear wave equation with the mixed nonhomogenous conditions: linear approximation and asymptotic expansion of solutions

Long N. T., Tam N. C., Truc N. T. T.

Demonstratio Mathematica

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2005

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Vol. 38, nr 2

365--386

EN

In this paper we consider the following nonlinear wave equation (1) utt-uxx = f(x,t,u,ux,ut), x is an element of (0,l), 0 < t < T, (2) ux(0,t)-hou(0,t)=go(t), u(1,t) = g1(t), (3) u(x,0) = uo(x), ut((x,0) = u1x).

14

Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms

Barbu V., Lasiecka I., Rammaha A.

Control and Cybernetics

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2005

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Vol. 34, no 3

665-687

EN

In this article we focus on the global well-posedness of the differential equation u [...] in Omega x(O, T), where j' denotes the derivative of a C1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with non-degenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obt ains existence, regularity or finite time blow up of solutions. More specifically, when p [is less than or equal to] m + k global existence of generalized solutions in H1 x L2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H2 x H1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.

15

Vibrations of an Elastic Body Carrying a Number of Concentrated Masses

Kasprzyk S., Sędziwy S.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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1999

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Vol. 47, nr 3

209-220

EN

The paper investigates the existence and uniqueness of the weak solution to the initial boundary value problem for a system consisting of a hyperbolic-type partial differential equation with distributional coefficients and a collection of ordinary differential equations, modelling small vibrations of a mechanical system composed with an elastic body and a number of concentrated masses.