scholarly journals Time-Dependent Neutral Stochastic Delay Partial Differential Equations Driven by Rosenblatt Process in Hilbert Space

2018 ◽  
pp. 301-320
Author(s):  
E. Lakhel ◽  
A. Tlidi
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Dependent ◽  
Partial Differential ◽  
Stochastic Delay ◽  
Rosenblatt Process ◽  
Delay Partial Differential Equations

scholarly journals Two-scale homogenization of abstract linear time-dependent PDEs

2020 ◽  
pp. 1-41
Author(s):  
Stefan Neukamm ◽  
Mario Varga ◽  
Marcus Waurick
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Linear Time ◽  
Random Coefficients ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Equations Of Mathematical Physics

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

ON EXPONENTIAL STABILITY FOR STOCHASTIC DELAY PARTIAL DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 09 (01) ◽  
pp. 121-134
Author(s):  
GUOSHENG YU ◽  
BING LIU
Keyword(s):  
Partial Differential Equations ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Exponential Stability ◽  
Hilbert Spaces ◽  
Stability Criteria ◽  
Partial Differential ◽  
Stochastic Delay ◽  
Finite Delay ◽  
Delay Partial Differential Equations

This paper is concerned with the exponential stability of energy solutions to a nonlinear stochastic delay partial differential equations with finite delay in separable Hilbert spaces. Some exponential stability criteria are obtained by constructing the Lyapunov function. As an application, one example is also given to illustrate our results.

scholarly journals The Existence, Uniqueness, and Controllability of Neutral Stochastic Delay Partial Differential Equations Driven by Standard Brownian Motion and Fractional Brownian Motion

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Dehao Ruan ◽  
Jiaowan Luo
Keyword(s):  
Fixed Point ◽  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Fixed Point Method ◽  
Standard Brownian Motion ◽  
Partial Differential ◽  
Stochastic Delay ◽  
Delay Partial Differential Equations

We focus on a class of neutral stochastic delay partial differential equations perturbed by a standard Brownian motion and a fractional Brownian motion. Under some suitable assumptions, the existence, uniqueness, and controllability results for these equations are investigated by means of the Banach fixed point method. Moreover, an example is presented to illustrate our main results.

Controllability for impulsive neutral stochastic delay partial differential equations driven by fBm and Lévy noise

2020 ◽  
Vol 21 (02) ◽  
pp. 2150013
Author(s):  
Diem Dang Huan ◽  
Ravi P. Agarwal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonlinear Heat Equation ◽  
Index Formula ◽  
Lévy Noise ◽  
Partial Differential ◽  
Stochastic Delay ◽  
Delay Partial Differential Equations ◽  
Levy Noise

This paper aims to investigate the controllability for impulsive neutral stochastic delay partial differential equations (PDEs) driven by fractional Brownian motion (fBm) with Hurst index [Formula: see text] and Lévy noise in Hilbert spaces. By using a fixed point approach without imposing a severe compactness condition on the semigroup, a new set of sufficient conditions is derived. The results in this paper are generalization and continuation of the recent results on this issue. At the end, an application to the stochastic nonlinear heat equation with delays driven by a fBm and Lévy noise is given.

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