Diffusion System Research Articles

Sampled-data controllers and observer-design for uncertain semilinear systems with positivity constraints

This paper deals with the problem of positive stabilization of a class of uncertain semilinear diffusion systems using sampled-data controllers. The conditions for the existence of such controllers are obtained through certain inequalities. By solving these inequalities, we determine upper bounds for the sampling intervals that guarantee exponential stability, as well as the corresponding decay rates. Furthermore, the design of a positive observer for systems with discrete measurements is studied. The stability of the estimation error is ensured through the same LMIs. Numerical simulations are presented to illustrate the effectiveness of the proposed approach.

Blow-up for a reaction–diffusion system with gradient terms under Robin boundary conditions

This paper is concerned with a reaction–diffusion system with gradient terms under Robin boundary conditions { ( h ( u ) ) t = ∇ ⋅ ( | ∇u | p − 2 ∇u ) + f ( u , v , | ∇u | 2 , t ) in Ω × ( 0 , t ∗ ) , ( r ( v ) ) t = ∇ ⋅ ( | ∇v | q − 2 ∇v ) + g ( u , v , | ∇v | 2 , t ) in Ω × ( 0 , t ∗ ) , ∂ u ∂ ν + γu = 0 , ∂ v ∂ ν + σv = 0 on ∂Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , v ( x , 0 ) = v 0 ( x ) ≥ 0 in Ω ¯ , where p, q>2 and Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with smooth boundary. Under some suitable assumptions, a sufficient condition that ensures the positive solution blows up in a finite time is obtained. Meanwhile, the upper bound for the blow-up time and the upper estimate for the blow-up rate are derived. The key to the research is to use the maximum principles and differential inequality technique. In addition, as an application of the abstract results obtained in this paper, an example is given.

Efficient uniformly convergent numerical methods for singularly perturbed parabolic reaction–diffusion systems with discontinuous source term

Efficient uniformly convergent numerical methods for singularly perturbed parabolic reaction–diffusion systems with discontinuous source term

A Dean–Kawasaki equation for reaction diffusion systems driven by Poisson noise

A Dean–Kawasaki equation for reaction diffusion systems driven by Poisson noise

Stability analysis of a conservative reaction–diffusion system with rate controls

Stability analysis of a conservative reaction–diffusion system with rate controls

Frictional effects on shear-induced diffusion in suspensions of non-Brownian particles

The shear-induced diffusivity of non-Brownian spheres in monodisperse suspensions undergoing viscous flow was calculated using simulations that account for particle roughness and friction as independent parameters. The diffusivity increases significantly as the friction coefficient is increased, and the effect is largest on rougher particles. Roughness reduces the transverse diffusivities relative to smoother particles for sufficiently concentrated suspensions of frictionless and low-friction particles. However, the diffusivity of roughened particles is larger than smoother ones at high values of the friction coefficient. The increase of the diffusivity with friction is associated with a significant broadening of the variance of the rotational velocities. The most prevalent observation, when correlating the microstructure to changes in diffusivity for frictionless particles, is that less diffusive systems, with larger roughness, form layers along the flow direction. These results confirm previous experimental and simulation results that roughness can decrease diffusivity at large concentrations using a more detailed model. Also, comparisons of the simulation results with previously published experimental measurements indicate that friction improves the alignment of the results with experiments.

Pattern Formation Mechanisms of Spatiotemporally Discrete Activator–Inhibitor Model with Self- and Cross-Diffusions

In this paper, we aim to solve the issue of pattern formation mechanisms in a spatiotemporally discrete activator–inhibitor model that incorporates self- and cross-diffusions. We seek to identify the conditions that lead to the emergence of complex patterns and to elucidate the principles governing the dynamic behaviors that result in these patterns. We first construct a corresponding coupled map lattice (CML) model based on the continuous activator–inhibitor reaction–diffusion system. In the reaction stage, we examine the existence, uniqueness, and stability of the homogeneous stationary state and specify the parametric conditions for realizing these properties. Furthermore, by applying the center manifold theorem, we perform a flip bifurcation analysis and confirm that the model is capable of undergoing flip bifurcation. In the diffusion stage, we focus on the analysis of Turing bifurcation and determine the parameter conditions for the emergence of Turing instability. Through numerical simulations, we validate and explain the results of our theoretical analysis. Our study reveals various Turing instability mechanisms by coupling Turing and flip bifurcations, which include pure-self-diffusion-Turing instability, pure-cross-diffusion-Turing instability, flip-self-diffusion-Turing instability, flip-cross-diffusion-Turing instability, and chaos-self-diffusion-Turing instability mechanisms. Under different mechanisms, we illustrate the corresponding Turing patterns and discover a rich variety of pattern types such as labyrinthine, mosaic, alternating mosaic, colorful mottled grid patterns with winding and twisted bands, and patterns with dense patches and twisted bands nested together. Our research provides a theoretical framework and numerical support for understanding the complex dynamical behaviors and pattern formations in activator–inhibitor models with self- and cross-diffusions.

Existence of similarity profiles for diffusion equations and systems

We study the existence of similarity profiles for diffusion equations and reaction diffusion systems on the real line, where the different nontrivial limits are imposed for x→-∞ and x→+∞. These similarity profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory.

Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities

AbstractIt is shown that semilinear parabolic evolution equations featuring Hölder continuous nonlinearities with at most linear growth possess global strong solutions for a general class of initial data. The abstract results are applied to a recent model describing front propagation in bushfires and in the context of a reaction–diffusion system.

The propagation dynamics for three species competitive-cooperative reaction–diffusion systems

The propagation dynamics for three species competitive-cooperative reaction–diffusion systems

Impact of Storage on In Vitro Permeation and Mucoadhesion Setup Experiments Using Swine Nasal Mucosa.

Intranasal topical administration offers a promising route for local and systemic drug delivery, with in vitro permeation and mucoadhesion studies often using porcine models. However, the impact of storage on mucosal integrity after the procedure remains unaddressed. This study aimed to standardize the preparation process and evaluated whether storage of porcine nasal mucosa impairs its integrity and permeability for experimental comparisons. Additionally, an optimized in vitro mucoadhesion experiment using texture analyzer equipment was investigated. Porcine nasal mucosa was subjected to different storage conditions ("fresh"; refrigerated at 4°C for 24h and 48h, and frozen at -20°C for two or three weeks) and assessed using optical and transmission electron microscopy. In vitro permeation assays were performed in a Franz-type vertical diffusion system with lidocaine hydrochloride (LDC). In vitro mucoadhesion assays were conducted using fresh nasal mucosa and a commercial nasal topical formulation using TA.XT. Plus texture analyzer. The variables involved (probe speed, contact time, and application force) in assessing mucoadhesive capacity (maximum mucoadhesive force Fmax and work of mucoadhesion Wmuc) were optimized using a Central Composite Design. Fresh tissues showed no alterations in histological arrangement or in the ultrastructure of adherence junctions. Stored tissues exhibited histological disorganization, reduced thickness, and loss of epithelial integrity. LDC permeability increased in storage tissues (p < 0.05). Contact force had a positive effect on Fmax and Wmuc (p < 0.0001), with a minimum required value of 0.48 N. Variations in contact time and probe speed did not affect the responses (p > 0.05). In conclusion, the preparation technique was adequate to maintain mucosa integrity for permeability studies. However, storing the mucosa at 4 or -20°C overestimated LDC permeation, which could mislead critical data for formulation development. Therefore, the use of fresh mucosa is recommended to ensure more reliable results. For in vitro mucoadhesion assays, a minimum contact force of 0.48N is required for optimal responses.

The role of spatial dimension in the emergence of localized radial patterns from a Turing instability

The emergence of localized radial patterns from a Turing instability has been well studied in two- and three-dimensional settings and predicted for higher spatial dimensions. We prove the existence of localized ( n + 1 ) -dimensional radial patterns in general two-component reaction–diffusion systems near a Turing instability, where n &gt; 0 is taken to be a continuous parameter. We determine the explicit dependence of each pattern’s radial profile on the dimension n through the introduction of ( n + 1 ) -dimensional Bessel functions, revealing a deep connection between the formation of localized radial patterns in different spatial dimensions.

Solar PV and clean cookstove Technology Diffusion Systems: Four case studies from Sub-Saharan Africa

Solar PV and clean cookstove Technology Diffusion Systems: Four case studies from Sub-Saharan Africa

Traveling wave solutions of a diffusive predator-prey system with Holling II type functional response

In this paper, we discuss a three-dimensional diffusive predator-prey system with nonlocal terms and Holling II type functional response. According to the relationship between traveling wave and heteroclinic orbit, the predator-prey system is transformed into the singularly perturbed system. Based on the method of the geometric singular perturbation theory, we construct a locally invariant manifold to obtain the traveling wave solutions with nonlocal delay convolution kernel.

Well-posedness, monotonicity, asymptotic translation of the Cauchy problem of delayed reaction–diffusion systems

In this paper, we establish a new basic theory on the Cauchy problem of delayed reaction–diffusion systems on the whole Euclidean space, where the initial function space is equipped with the compact open topology (also called coarse topology). Generally, under this coarse topology, the reaction terms are not locally Lipschitz continuous. Even they are, it is difficult to boil down the basic theory on the Cauchy problem to the (parameterized) contraction mapping problem. To overcome this difficulty, we explain and prove the uniqueness of solutions from a new perspective. This provides a unified way to obtain the basic theory on the Cauchy problem of delayed reaction–diffusion systems on the whole Euclidean space, which includes the local existence, uniqueness, and continuous dependence of solutions. Moreover, we show that, with respect to the compact open topology, the solution semiflow is monotone and possesses the property of asymptotic translation.

Sensing Applications of PT‐Symmetry in Non‐Hermitian Photonic Systems

AbstractIn recent years, rapid advances in non‐Hermitian physics and PT‐symmetry have brought new opportunities for ultra‐sensitive sensing. Especially the presence of controllable non‐conservative processes in optical and photonic systems has triggered the development of singularity‐based sensing. By flexibly tuning gain, loss, and coupling strength, a series of high‐resolution sensing approaches can be realized, with the potential of on‐chip integration. Another important non‐Hermitian singularity is the coherent perfect absorption‐lasing (CPAL) point in the PT‐broken phase, which manifests the coexistence of lasing and CPA, exhibiting intriguing properties with considerable sensing potential. As a crucial method for quantum sensing and metrology, the interaction between light and alkali‐metal atomic ensembles promises unprecedented sensitivity in the measurement of ultra‐weak magnetic field, inertia, and time. Therefore, extending the study of PT‐symmetry and singularity‐based sensing from conventional solid‐state wave systems to diffusive systems such as atomic ensembles is attracting wide attention. In this review, the development of singularity‐based sensing in PT/anti‐PT symmetric non‐Hermitian systems is summarized, with a special focus on photonic platforms including integration with waveguides, microcavities, metasurface, etc. In addition, sensing applications with discussion further extended to atomic ensembles, projecting future research trends in the field.

Random attractors for fractional stochastic reaction–diffusion systems with fractional Brownian motion

Fractional stochastic reaction–diffusion systems involving fractional Brownian motion (fBm) provide a comprehensive modeling framework for studying complex physical and biological phenomena. In this study, we investigate the dynamic behavior of solutions to fractional stochastic reaction–diffusion systems with fBm defined on Rn. Firstly, we establish the existence and uniqueness of solutions to the fractional stochastic reaction–diffusion systems with fBm. We also obtain uniform estimates for the solutions on average. Furthermore, we construct a mean random dynamical system based on the derived solutions. Finally, we prove the existence and uniqueness of weak pullback mean random attractors. The results of our investigation contribute to a deeper understanding of the complexities involved in reaction–diffusion processes, specifically considering the effect of fBm with long-range dependence and self-similarity properties. Additionally, our findings have broader applications, particularly in areas such as analyzing the dynamic behavior of complex systems and biological models.

Finite-Time Stability Analysis of a Discrete-Time Generalized Reaction–Diffusion System

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab’s LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems.

Propagation dynamics of the lattice Leslie-Gower predator-prey system in shifting habitats

In this paper, we are concerned with the propagation dynamics of a discrete diffusive Leslie-Gower predator-prey system in shifting habitats. First, we discuss the spreading properties of the corresponding Cauchy problem depending on the range of the shifting speed which is identified respectively by (i) extinction of two species; (ii) only one species surviving; (iii) persistence of two species. Then, we give the existence of two types of forced waves, that is, I type forced waves invading the state where only one species exists in supercritical case and critical case, and II type forced waves invading coexistence state for any speed.

Global well-posedness and uniform boundedness of 2D urban crime models with nonlinear advection enhancement

Abstract We study the global well-posedness and uniform boundedness of a two-dimensional reaction–advection–diffusion system with nonlinear advection. This strongly coupled system of nonlinear partial differential equations represents the continuum of a 2D lattice model designed to describe residential burglary, where each location is characterised by a tractability value that varies in both space and time. We show that the model with sublinear advection enhancement is globally well-posed, with a unique solution that is classical and uniformly bounded in time. Our results provide valuable insights into the development of urban crime models with nonlinear advection enhancements, making them suitable for broader applications, including nonlocal or heterogeneous near-repeat victimisation effects.