scholarly journals Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Xuewei Ju ◽  
Hongli Wang ◽  
Desheng Li ◽  
Jinqiao Duan
Keyword(s):  
Boundary Value Problem ◽  
White Noise ◽  
Global Attractor ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Boundary Value ◽  
Uniqueness Result ◽  
Hilliard Equation ◽  
Bounded Set ◽  
Cahn Hilliard Equation

This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.

STOCHASTIC CAHN–HILLIARD EQUATION WITH FRACTIONAL NOISE

2008 ◽  
Vol 08 (04) ◽  
pp. 643-665 ◽  
Author(s):  
LIJUN BO ◽  
YIMING JIANG ◽  
YONGJIN WANG
Keyword(s):  
Weak Convergence ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Fractional Noise ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Fractional Noises

We study the existence and uniqueness of global mild solutions to a class of stochastic Cahn–Hilliard equations driven by fractional noises (fractional in time and white in space), through a weak convergence argument.

On Cahn—Hilliard systems with elasticity

2003 ◽  
Vol 133 (2) ◽  
pp. 307-331 ◽  
Author(s):  
Harald Garcke
Keyword(s):  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Separation Process ◽  
Diffusion Equations ◽  
System Of Equations ◽  
Ginzburg Landau ◽  
Elastic Interactions ◽  
Hilliard Equation ◽  
Phase Separation Process ◽  
Cahn Hilliard Equation

Elastic effects can have a pronounced effect on the phase-separation process in solids. The classical Ginzburg—Landau energy can be modified to account for such elastic interactions. The evolution of the system is then governed by diffusion equations for the concentrations of the alloy components and by a quasi-static equilibrium for the mechanical part. The resulting system of equations is elliptic-parabolic and can be understood as a generalization of the Cahn—Hilliard equation. In this paper we give a derivation of the system and prove an existence and uniqueness result for it.

Fixed point results for single valued and set valued P-contractions and application to second order boundary value problems

2020 ◽  
Vol 36 (2) ◽  
pp. 205-214
Author(s):  
ISHAK ALTUN ◽  
HATICE ASLAN HANCER ◽  
ALI ERDURAN ◽  
◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Boundary Value ◽  
Uniqueness Result ◽  
Second Order ◽  
Point Boundary ◽  
Set Valued Mappings

In this paper, by considering the concept of set-valued nonlinear P-contraction which is newly introduced, we present some new fixed point theorems for set-valued mappings on complete metric space. Then by considering a single-valued case we provide an existence and uniqueness result for a kind of second order two point boundary value problem.

Global attractor for a hyperbolic Cahn–Hilliard equation with a proliferation term

2021 ◽  
pp. 1-23
Author(s):  
Padouette Boubati Badieti Matala ◽  
Daniel Moukoko ◽  
Mayeul Evrard Isseret Goyaud
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Global Attractor ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Hilliard Equation ◽  
Uniqueness Of The Solution ◽  
Cahn Hilliard Equation

In this article, we study a hyperbolic equation of Cahn–Hilliard with a proliferation term and Dirichlet boundary conditions. In particular, we prove the existence and uniqueness of the solution, and also the existence of the global attractor.

scholarly journals On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative

2019 ◽  
Vol 39 (2) ◽  
pp. 131-144
Author(s):  
Michał Bełdziński ◽  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Unique Solvability ◽  
Existence And Uniqueness ◽  
Variational Approach ◽  
Boundary Value ◽  
Uniqueness Result ◽  
Second Order

In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.

scholarly journals Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuqi Wang ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Novel Approach ◽  
Banach Contraction ◽  
Left And Right ◽  
Banach Contraction Mapping Principle

AbstractIn this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals An extension of Gronwall inequality in the theory of bodies with voids

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1161-1167
Author(s):  
Marin Marin ◽  
Praveen Ailawalia ◽  
Ioan Tuns
Keyword(s):  
Boundary Value Problem ◽  
Internal State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Internal Variables ◽  
State Variables ◽  
Gronwall’S Inequality ◽  
Initial Boundary Value

Abstract In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

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