scholarly journals Abstract semilinear stochastic Itó-Volterra integrodifferential equations

2006 ◽  
Vol 2006 ◽  
pp. 1-22 ◽  
Author(s):  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Existence And Uniqueness ◽  
Separable Hilbert Space ◽  
Analogous Result ◽  
Growth Conditions ◽  
Integrodifferential Equations ◽  
Time Dependent ◽  
Abstract Theory ◽  
Approximation Result ◽  
Global Existence And Uniqueness ◽  
Real Separable Hilbert Space

We consider a class of abstract semilinear stochastic Volterra integrodifferential equations in a real separable Hilbert space. The global existence and uniqueness of a mild solution, as well as a perturbation result, are established under the so-called Caratheodory growth conditions on the nonlinearities. An approximation result is then established, followed by an analogous result concerning a so-called McKean-Vlasov integrodifferential equation, and then a brief commentary on the extension of the main results to the time-dependent case. The paper ends with a discussion of some concrete examples to illustrate the abstract theory.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR A SEMICONDUCTOR DRIFT-DIFFUSION-POISSON MODEL

2008 ◽  
Vol 18 (03) ◽  
pp. 443-487 ◽  
Author(s):  
HAO WU ◽  
PETER A. MARKOWICH ◽  
SONGMU ZHENG
Keyword(s):  
Global Existence ◽  
Existence And Uniqueness ◽  
Poisson Model ◽  
Stationary Problem ◽  
Time Dependent ◽  
Unique Equilibrium ◽  
Poisson System ◽  
Drift Diffusion ◽  
Global Existence And Uniqueness ◽  
Time Dependent Problem

In this paper a time-dependent as well as a stationary drift-diffusion-Poisson system for semiconductors are studied. Global existence and uniqueness of weak solution of the time-dependent problem are proven and we also prove the existence and uniqueness of the steady state. It is shown that as time tends to infinity, the solution of the time-dependent problem will converge to a unique equilibrium. Due to the presence of recombination-generation rate R in our drift-diffusion-Poisson model, the work of this paper in some sense extends the results in the previous literature (on both time-dependent problem and stationary problem).

Complete Controllability of Fractional Impulsive Multivalued Stochastic Partial Integrodifferential Equations with State-Dependent Delay

2017 ◽  
Vol 18 (3-4) ◽  
pp. 197-220 ◽  
Author(s):  
Zuomao Yan ◽  
Fangxia Lu
Keyword(s):  
Separable Hilbert Space ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Complete Controllability ◽  
Approximation Techniques ◽  
State Dependent ◽  
State Dependent Delay ◽  
Multivalued Operators ◽  
Real Separable Hilbert Space ◽  
The Fixed Point Theorem

AbstractIn this article, we consider a class of fractional impulsive multivalued stochastic partial integrodifferential equations with state-dependent delay in a real separable Hilbert space. Sufficient conditions for the complete controllability of impulsive fractional stochastic evolution systems are established by means of the fixed-point theorem for discontinuous multivalued operators due to Dhage and properties of the $\alpha$-resolvent operator combined with approximation techniques. Two examples are also given to illustrate the obtained theorem.

scholarly journals Attractors for Nonautonomous Multivalued Evolution Systems Generated by Time-Dependent Subdifferentials

2002 ◽  
Vol 7 (9) ◽  
pp. 453-473 ◽  
Author(s):  
Noriaki Yamazaki
Keyword(s):  
Dynamical Systems ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Global Attractors ◽  
Time Dependent ◽  
Initial State ◽  
Multivalued Dynamical Systems ◽  
Real Separable Hilbert Space ◽  
The Relationship ◽  
Multivalued Dynamical System

In a real separable Hilbert space, we consider nonautonomous evolution equations including time-dependent subdifferentials and their nonmonotone multivalued perturbations. In this paper, we treat the multivalued dynamical systems associated with time-dependent subdifferentials, in which the solution is not unique for a given initial state. In particular, we discuss the asymptotic behaviour of our multivalued semiflows from the viewpoint of attractors. In fact, assuming that the time-dependent subdifferential converges asymptotically to a time-independent one (in a sense) as time goes to infinity, we construct global attractors for nonautonomous multivalued dynamical systems and its limiting autonomous multivalued dynamical system. Moreover, we discuss the relationship between them.

scholarly journals On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

2007 ◽  
Vol 2007 ◽  
pp. 1-26 ◽  
Author(s):  
Eduardo Hernandez ◽  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Real Separable Hilbert Space

We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.

scholarly journals Controllability for neutral stochastic functional integrodifferential equations with infinite delay

2016 ◽  
Vol 1 (2) ◽  
pp. 493-506 ◽  
Author(s):  
Tomás Caraballo ◽  
Mamadou Abdoul Diop ◽  
Aziz Mane
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Separable Hilbert Space ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Resolvent Operators ◽  
Real Separable Hilbert Space ◽  
Stochastic Functional

AbstractIn this work, we study the controllability for a class of nonlinear neutral stochastic functional integrodifferential equations with infinite delay in a real separable Hilbert space. Sufficient conditions for the controllability are established by using Nussbaum fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result.

scholarly journals Functional integro-differential stochastic evolution equations in Hilbert space

2003 ◽  
Vol 16 (2) ◽  
pp. 141-161 ◽  
Author(s):  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Mathematical Modeling ◽  
Hilbert Space ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Existence Results ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Abstract Theory ◽  
Convergence Results ◽  
Real Separable Hilbert Space

We investigate a class of abstract functional integro-differential stochastic evolution equations in a real separable Hilbert space. Global existence results concerning mild and periodic solutions are formulated under various growth and compactness conditions. Also, related convergence results are established and an example arising in the mathematical modeling of heat conduction is discussed to illustrate the abstract theory.

Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

2021 ◽  
pp. 1-18
Author(s):  
Yuming Qin ◽  
Bin Yang
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Force Term ◽  
Pullback Attractors ◽  
Critical Exponential Growth ◽  
Nonclassical Diffusion Equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

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