scholarly journals Stochastic Chemotaxis Model with Fractional Derivative Driven by Multiplicative Noise

2021 ◽  
Vol 19 (6) ◽  
pp. 858-889
Author(s):  
Ali Slimani ◽  
Amira Rahai ◽  
Amar Guesmia ◽  
Lamine Bouzettouta
Keyword(s):  
Fractional Derivative ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Multiplicative Noise ◽  
Dirichlet Boundary ◽  
Semigroup Theory ◽  
Dirichlet Boundary Conditions ◽  
Chemotaxis Model ◽  
Analysis Techniques ◽  
Regularity Properties

We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deterministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study of nonlinear stochastic chemotaxis model with Dirichlet boundary conditions, fractional Derivative and disturbed by multiplicative noise. The required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model.

scholarly journals Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Fathalla A. Rihan ◽  
Chinnathambi Rajivganthi ◽  
Palanisamy Muthukumar
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Poisson Jumps ◽  
Hilfer Fractional Derivative ◽  
Analysis Techniques ◽  
Mild Conditions ◽  
Fractional Stochastic Differential Equations ◽  
Existence Of Optimal Control

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.

scholarly journals On a 2D stochastic Euler equation of transport type: Existence and geometric formulation

2014 ◽  
Vol 15 (01) ◽  
pp. 1450012 ◽  
Author(s):  
Ana Bela Cruzeiro ◽  
Iván Torrecilla
Keyword(s):  
Boundary Conditions ◽  
Euler Equation ◽  
Multiplicative Noise ◽  
Dirichlet Boundary ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Geometric Formulation ◽  
Transport Type

We prove weak existence of Euler equation (or Navier–Stokes equation) perturbed by a multiplicative noise on bounded domains of ℝ2 with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are H1 regular. The equations are of transport type.

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

A Lane–Emden system with singular data

2011 ◽  
Vol 141 (6) ◽  
pp. 1279-1294 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Singular Data

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

scholarly journals On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Le Thi Phuong Ngoc ◽  
Khong Thi Thao Uyen ◽  
Nguyen Huu Nhan ◽  
Nguyen Thanh Long
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Local Existence ◽  
Initial Energy ◽  
Dirichlet Boundary Conditions ◽  
Local Existence And Uniqueness ◽  
Pseudoparabolic Equations

<p style='text-indent:20px;'>In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.</p>

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO 2D G-BÉNARD PROBLEM IN UNBOUNDED DOMAINS

2020 ◽  
Vol 65 (6) ◽  
pp. 23-30
Author(s):  
Thinh Tran Quang ◽  
Thuy Le Thi
Keyword(s):  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Uniqueness Of Solutions ◽  
Navier Stokes Equations ◽  
Bénard Problem ◽  
Benard Problem ◽  
Homogeneous Dirichlet Boundary Conditions

We consider the 2D g-Bénard problem in domains satisfying the Poincaré inequality with homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of global weak solutions. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Bénard 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 Existence and Uniqueness of Abstract Stochastic Fractional-Order Differential Equation

2019 ◽  
Vol 16 ◽  
pp. 8280-8287
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
A. Tarek S.A.
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Stochastic Analysis ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Analysis Techniques ◽  
Fractional Order Differential Equations

In this paper, the existence and uniqueness about the solution for a class of abstract stochastic fractional-order differential equations                                           where  in and  are given functions, are investigated, where the fractional derivative is described in Caputo sense. The fractional calculus, stochastic analysis techniques and the standard $Picard's$ iteration method are used to obtain the required.

scholarly journals On weak solutions of semilinear hyperbolic-parabolic equations

1996 ◽  
Vol 19 (4) ◽  
pp. 751-758 ◽  
Author(s):  
Jorge Ferreira
Keyword(s):  
Parabolic Equation ◽  
Continuous Function ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
A Priori ◽  
A Priori Estimate ◽  
Dirichlet Boundary Conditions ◽  
Priori Estimate

In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation(K1(x,t)u′)′+K2(x,t)u′+A(t)u+F(u)=fwith null Dirichlet boundary conditions and zero initial data, whereF(s)is a continuous function such thatsF(s)≥0,∀s∈Rand{A(t);t≥0}is a family of operators ofL(H01(Ω);H−1(Ω)). For the existence we apply the Faedo-Galerkin method with an unusual a priori estimate and a result of W. A. Strauss. Uniqueness is proved only for some particular classes of functionsF.

Monotone iterative sequences for non-local elliptic problems

2011 ◽  
Vol 22 (6) ◽  
pp. 533-552 ◽  
Author(s):  
MOHAMMED AL-REFAI ◽  
NIKOS I. KAVALLARIS ◽  
MOHAMED ALI HAJJI
Keyword(s):  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Uniqueness Result ◽  
Dirichlet Boundary Conditions ◽  
Iterative Sequence ◽  
Monotone Sequence ◽  
Monotone Iterative ◽  
Elliptic Differential Equations ◽  
Actual Solution ◽  
Non Local

In this paper we establish an existence and uniqueness result for a class of non-local elliptic differential equations with the Dirichlet boundary conditions, which, in general, do not accept a maximum principle. We introduce one monotone sequence of lower–upper pairs of solutions and prove uniform convergence of that sequence to the actual solution of the problem, which is the unique solution for some range of λ (the parameter of the problem). The convergence of the iterative sequence is tested through examples with an order of convergence greater than 1.

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