Stochastic Reaction-diffusion Research Articles

Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains

This article is concerned with the limiting behavior of dynamics of a class of non-autonomous stochastic partial differential equations driven by colored noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on $ (n+1) $-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the $ (n+1) $-dimensional unbounded thin domains collapse onto the $ n $-dimensional space $ \mathbb{R}^n $. Here, the tail estimates are utilized to deal with the non-compactness of Sobolev embeddings on unbounded domains.

Optimal control of stochastic phase-field models related to tumor growth

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

Averaging principle for slow–fast stochastic Burgers equation driven by [formula omitted]-stable process

In this paper, we study the averaging principle for a slow–fast stochastic system driven by α-stable process with α∈(1,2), where the slow and fast components are the stochastic Burger equation and the stochastic reaction–diffusion equation respectively. The proof is based on the Khasminskii’s time discretization and the techniques of stopping times.

Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.

Tempered random attractors of a non-autonomous non-local fractional equation driven by multiplicative white noise

In this paper, we prove the existence and regularity of a tempered pullback attractor for a non-autonomous stochastic fractional equation involving an abstract non-local operator defined aswhere is the kernel of which satisfies the general fractional-type condition of order and is a ball in centered at x with radius ε. The asymptotical compactness of solutions in is demonstrated by the spectral decomposition and truncation technique, where is a Hilbert space with and As a special example, we obtain the existence and regularity of tempered pullback attractor of stochastic fractional reaction-diffusion equation.

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by $\alpha$-stable process with $\alpha\in (1,2)$. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.

Invariant measures for stochastic reaction–diffusion equations with weakly dissipative nonlinearities

ABSTRACT In this paper we establish the existence of an invariant measure for a stochastic reaction–diffusion equation of the type , where f and σ are nonlinear maps and W is an infinite dimensional Q-–Wiener process. Our emphasis is on unbounded domain . Under a very mild dissipation assumption, we show the existence of a solution which is bounded in probability using a general Ito's formula. In addition, we investigate a type of equation for which bounded solutions may be obtained as a limit of an iteration scheme. Together with the compactness property of the heat equation semigroup, stochastic continuity and Feller property of the transition semigroup, the existence of such solution implies the existence of an invariant measure which is an important step in establishing the ergodic behaviour of the underlying physical system.

Extended dissipativity stabilization and synchronization of uncertain stochastic reaction-diffusion neural networks via intermittent non-fragile control

This paper mainly studies the problem of extended dissipativity stabilization and synchronization of a class of uncertain stochastic reaction-diffusion neural networks with discrete and distributed time-varying delays via intermittent non-fragile control strategies. A novel stabilization and synchronization criteria and extended dissipative analysis is obtained by combining a switching time dependent Lyapunov functional method with Wirtinger’s inequality technique. Then, by solving a set of delay dependent linear matrix inequalities, we obtain the desired intermittent non-fragile controller, which can be used to satisfies the prescribed level of extended dissipativity behaviors, including H∞ behavior, passivity action, (Q, S, R)-dissipative performance, and L2−L∞ performance. Finally, two numerical examples are adopted to verify the validity of the stabilization and synchronization results.

Stability of Impulsive Stochastic Reaction Diffusion Recurrent Neural Network

In this paper, the problem of global asymptotic stability of stochastic Markovian jumping reaction-diffusion neural networks with discrete and distributed delays is investigated. By utilizing the Lyapunov–Krasovskii functional method combined with linear matrix inequality approach, novel sufficient stability conditions are derived for impulsive stochastic reaction-diffusion recurrent neural networks with Markovian jumping parameters and mixed delays. Finally, numerical examples with simulation results are given to illustrate the derived theoretical results.

Turnover versus treadmilling in actin network assembly and remodeling.

Actin networks are highly dynamic cytoskeletal structures that continuously undergo structural remodeling. One prominent way to probe these processes is via Fluorescence Recovery After Photobleaching (FRAP), which can be used to estimate the rate of turnover for filamentous actin monomers. It is thought that head-to-tail treadmilling and de novo filament nucleation constitute two primary mechanisms underlying turnover kinetics. More generally, these self-assembly activities are responsible for many important cellular functions such as force generation, cellular shape dynamics, and cellular motility. In what relative proportions filament treadmilling and de novo filament nucleation contribute to actin network turnover is still not fully understood. We used an advanced stochastic reaction-diffusion model in three dimensions, MEDYAN, to study turnover dynamics of actin networks containing Arp2/3, formin and capping protein at experimentally meaningful length- and time-scales. Our results reveal that, most commonly, treadmilling of older filaments is the main contributor to actin network turnover. On the other hand, although turnover and treadmilling are often used interchangeably, we show clear instances where this assumption would not be justified, for example, finding that rapid turnover is accompanied by slow treadmilling in highly dendritic Arp2/3 networks.

The Influence of Molecular Reach and Diffusivity on the Efficacy of Membrane-Confined Reactions

Signaling by surface receptors often relies on tethered reactions whereby an enzyme bound to the cytoplasmic tail of a receptor catalyzes reactions on substrates within reach. The overall length and stiffness of the receptor tail, the enzyme, and the substrate determine a biophysical parameter termed the molecular reach of the reaction. This parameter determines the probability that the receptor-tethered enzyme will contact the substrate in the volume proximal to the membrane when separated by different distances within the membrane plane. In this work, we develop particle-based stochastic reaction-diffusion models to study the interplay between molecular reach and diffusion. We find that increasing the molecular reach can increase reaction efficacy for slowly diffusing receptors, whereas for rapidly diffusing receptors, increasing molecular reach reduces reaction efficacy. In contrast, if reactions are forced to take place within the two-dimensional plasma membrane instead of the three-dimensional volume proximal to it or if molecules diffuse in three dimensions, increasing molecular reach increases reaction efficacy for all diffusivities. We show results in the context of immune checkpoint receptors (PD-1 dephosphorylating CD28), a standard opposing kinase-phosphatase reaction, and a minimal two-particle model. The work highlights the importance of the three-dimensional nature of many two-dimensional membrane-confined interactions, illustrating a role for molecular reach in controlling biochemical reactions.

Stabilisation of stochastic delay Markovian reaction‐diffusion systems via boundary control

This study considers asymptotic stability in the mean square sense for stochastic delay Markovian reaction-diffusion systems (SDMRDSs) via boundary control. Firstly, the authors present a boundary controller for the system. By constructing of a Lyapunov–Krasovskii functional and utilising of Poincaré inequality, a sufficient criterion of mean square asymptotic stability for SDMRDSs is established. Next, boundary control for stochastic Markovian reaction-diffusion systems (SMRDSs) with time-varying delays is also investigated, and a delay-dependent sufficient condition for SMRDSs under the boundary control law is obtained. In addition, the robust boundary stabilisation of SMRDSs with parametric uncertainty is considered. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed results.

Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise

This paper is concerned with the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on Rn. We first prove the well-posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors. We finally establish the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains.

Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains

This article is concerned with the limiting behavior of dynamics of a class of nonautonomous stochastic partial differential equations driven by multiplicative white noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on (n + 1)-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the (n + 1)-dimensional unbounded thin domains collapse onto the n-dimensional space Rn. Here, the tail estimates are utilized to deal with the noncompactness of Sobolev embeddings on unbounded domains.

Resilience for stochastic systems interacting via a quasi-degenerate network.

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process, which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.

Turing-like mechanism in a stochastic reaction-diffusion model recreates three dimensional vascular patterning of plant stems.

Vascular tissue in plants provides a resource distribution network for water and nutrients that exhibits remarkable diversity in patterning among different species. In many succulent plants, the vascular network includes longitudinally-oriented supplemental vascular bundles (SVBs) in the central core of the succulent stems and roots in addition to the more typical zone of vascular tissue development (vascular cambium) in a cylinder at the periphery of the succulent organ. Plant SVBs evolved in over 38 plant families often in tandem with evolutionary increases in stem and root parenchyma storage tissue, so it is of interest to understand the evolutionary-developmental processes responsible for their recurrent evolution and patterning. Previous mathematical models have successfully recreated the two-dimensional vascular patterns in stem and root cross sections, but such models have yet to recreate three-dimensional vascular patterning. Here, a stochastic reaction-diffusion model of plant vascular bundle patterning is developed in an effort to highlight a potential mechanism of three dimensional patterning–Turing pattern formation coupled with longitudinal efflux of a regulatory molecule. A relatively simple model of four or five molecules recreated empirical SVB patterns and many other common vascular arrangements. SVBs failed to develop below a threshold width of parenchymatous tissues, suggesting a mechanism of evolutionary character loss due to changes in the spatial context in which development takes place. Altered diffusion rates of the modeled activator and substrate molecules affected the number and size of the simulated SVBs. This work provides a first mathematical model employing a stochastic Turing-type mechanism that recreates three dimensional vascular patterns seen in plant stems. The model offers predictions that can be tested using molecular-genetic approaches. Evolutionary-developmental ramifications concerning evolution of diffusion rates, organ size and geometry are discussed.

Averaging principle for Korteweg–de Vries equation with a random fast oscillation

This work concerns averaging principle for Korteweg–de Vries equation perturbed by a oscillating term which arises from the solution of a stochastic reaction–diffusion equation. This model can be translated into multiscale stochastic partial differential equations. Under some suitable conditions, we show that the slow component strongly converges to the solution of a single Korteweg–de Vries equation with a modified coefficient (averaged equation). To be more precise, by using the Khasminskii technique, we can obtain the strong convergence rate of the slow component towards the solution of the corresponding averaged equation, and as a consequence, the multiscale system can be reduced to the averaged equation.

Limiting dynamics for stochastic reaction–diffusion equations on the Sobolev space with thin domains

This paper is devoted to bi-spatial random attractors of the stochastic reaction–diffusion equation when the terminate space is the Sobolev space on a thin domain, where the nonlinearity can be decomposed into two functions with (p,q)-growth exponents. By means of a computation method of induction, it is shown that the difference of solutions near the initial time is integrable of kp−2k+2 order. This higher-order integrability shows continuity of the solution operator from the square Lebesgue space to (kp−2k+2)-order Lebesgue spaces. In particular, the (2p−2)-order integrability shows continuity of the solution operator from the square Lebesgue space to the Sobolev space, which further shows existence of a random attractor in the Sobolev space when the initial space is the square Lebesgue space. Moreover, the higher-order integrability can be uniform with respect to all thin domains, which provides uniformly asymptotic compactness of the random dynamical systems. As a conclusion, the upper semi-continuity of attractors under the Sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain.

Higher-order Wong–Zakai approximations of stochastic reaction–diffusion equations on [formula omitted

In this paper, we consider the higher-order Wong–Zakai approximations of the non-autonomous stochastic reaction–diffusion equation driven by additive/multiplicative white noises. The solutions between the approximation equation and stochastic reaction–diffusion equation are compared in higher-order spaces, in terms of the initial data. Based on these results and the known L2-upper semi-continuity, we prove that the random attractor of the approximation random system converges to that of the non-autonomous stochastic reaction–diffusion equation with additive/multiplicative white noises in Lp(RN)∩H1(RN) when the size of the approximation shrinks to zero.

Stability analysis of stochastic BAM neural networks with reaction–diffusion, multi-proportional and distributed delays

This paper is devoted to investigating of the stability for stochastic reaction–diffusion BAM neural networks with mixed delays. By applying some new analysis methods, several novel exponential stability criteria are obtained. Our results extend some existing results on stochastic BAM neural networks including with/without reaction–diffusion, time-varying (TV) and multi-proportional delays. In particular, we consider the effect of TV, distributed and multi-proportional delays. An example is provided to show the effectiveness of the obtained results.