scholarly journals Existence and Stability Estimate for the Solution of the Ageing Hereditary Linear Viscoelasticity Problem

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Julia Orlik
Keyword(s):  
Boundary Conditions ◽  
Viscoelastic Material ◽  
Weak Solutions ◽  
Linear Viscoelasticity ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Quasistatic Evolution ◽  
Variational Solutions ◽  
Existence And Stability ◽  
Viscoelasticity Problem

The paper is concerned with the existence and stability of weak (variational) solutions for the problem of the quasistatic evolution of a viscoelastic material under mixed inhomogenous Dirichlet-Neumann boundary conditions. The main novelty of the paper relies in dealing with continuous-in-time weak solutions and allowing nonconvolution and weak-singular Volterra's relaxation kernels.

Inverse problem for a degenerate/singular parabolic system with Neumann boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed Alaoui ◽  
Abdelkarim Hajjaj ◽  
Lahcen Maniar ◽  
Jawad Salhi
Keyword(s):  
Boundary Conditions ◽  
Parabolic System ◽  
Source Term ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Carleman Estimates ◽  
Well Posedness ◽  
Degenerate Diffusion ◽  
Neumann Type ◽  
Lower Order Terms

AbstractIn this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique.

scholarly journals Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Ghasem A. Afrouzi ◽  
Z. Naghizadeh ◽  
Nguyen Thanh Chung
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Weak Solutions ◽  
Multiple Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Nonlocal Problems ◽  
Mapping Theory ◽  
Nonlinear Neumann Boundary Conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

scholarly journals Solutions to a model with Neumann boundary conditions for sea-ice growth

2021 ◽  
Author(s):  
Yangxin Tang ◽  
Lin Zheng
Keyword(s):  
Global Existence ◽  
Sea Ice ◽  
Boundary Conditions ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Dirichlet Boundary Conditions ◽  
Time Behavior ◽  
Ice Growth ◽  
Long Time

We continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea-ice growth. In a previous paper global existence and the long-time of behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here we show that the global existence of weak solutions and the long-time behavior are also studied under Neumann boundary condition. In this paper we study in space dimension lower than or equal to $3$.

scholarly journals Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

2020 ◽  
Vol 6 (2) ◽  
pp. 685-709
Author(s):  
Gurusamy Arumugam ◽  
André H. Erhardt
Keyword(s):  
Neumann Problem ◽  
Weak Solutions ◽  
Type Inequality ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Energy Estimates ◽  
Variable Exponents ◽  
Existence Of Weak Solutions ◽  
Cross Diffusion ◽  
Time Variables

Abstract This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

scholarly journals Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

2015 ◽  
Vol 17 (03) ◽  
pp. 1450016
Author(s):  
Futoshi Takahashi
Keyword(s):  
Boundary Conditions ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Extremal Solutions ◽  
Smooth Bounded Domain ◽  
Suitable Notion ◽  
Extremal Parameter ◽  
Nonlinear Neumann Boundary Conditions ◽  
Increasing Function

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions [Formula: see text] where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Study of fluid flow through a channel deformed by piezoelement

2018 ◽  
Vol 13 (3) ◽  
pp. 1-10 ◽  
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh Nasibullaeva ◽  
O.V. Darintsev
Keyword(s):  
Fluid Flow ◽  
Pressure Drop ◽  
Boundary Conditions ◽  
Flow Rate ◽  
Contact Surface ◽  
Piezoelectric Element ◽  
Neumann Boundary ◽  
Differential Pressure ◽  
Physical Parameters ◽  
Flow Through

The flow of a liquid through a tube deformed by a piezoelectric cell under a harmonic law is studied in this paper. Linear deformations are compared for the Dirichlet and Neumann boundary conditions on the contact surface of the tube and piezoelectric element. The flow of fluid through a deformed channel for two flow regimes is investigated: in a tube with one closed end due to deformation of the tube; for a tube with two open ends due to deformation of the tube and the differential pressure applied to the channel. The flow rate of the liquid is calculated as a function of the frequency of the deformations, the pressure drop and the physical parameters of the liquid.

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