Homogeneous Conditions Research Articles

Existence results for Cahn–Hilliard-type systems driven by nonlocal integrodifferential operators with singular kernels

We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as minimizing movements scheme, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the q-Laplacian.In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.

Temperature imaging during the hydrogen release reaction from a liquid organic hydrogen carrier (LOHC) system using phosphor thermometry

Liquid organic hydrogen carrier (LOHC) systems offer a particularly interesting option for chemical hydrogen storage. In order to characterize and understand the endothermal hydrogen release from the carrier liquid and to evaluate suitable catalyst materials, knowledge of the temperature fields in the dehydrogenation reactor is important. One suitable technique for planar temperature sensing in reacting systems is phosphor thermometry. It is based on the excitation of a luminescent material by a laser pulse and detection of the subsequent phosphorescence signal. We investigated the luminescence of the thermographic phosphor (Sr,Ca)SiAIN3:Eu2+ (“SCASN:Eu2+”) dispersed in the H0-DBT / H18-DBT LOHC system in a temperature range from 400 to 600 K. A measurement cell enables repeatable and homogeneous measurement conditions of the hydrogen release reaction. A catalytic plate was put inside the heated LOHC. Temperature fields during the hydrogen release reaction were measured for the first time using the phosphorescence decay time (PDT) and the phosphorescence intensity ratio method (PIR). As expected, a strong cooling at the catalyst surface during the endothermal hydrogen release reaction could be observed, which was quantified to be in the range of 40 K.Graphical abstract

Global asymptotic stability in a two-dimensional chemotaxis model arising from tumor invasion

This paper considers the chemotaxis model with density-suppressed motility: ut = ∇·(φ(v)∇u) + ∇·(ψ(v)u∇v) + f(u), vt = Δv + wz, wt = −wz, wt = −wz, zt = Δz − z + u, x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2. Given that the positive motility function φ(v) has the lower-upper bound, we can conclude that the system possesses a unique bounded classical solution. Moreover, it is proved that the global bounded solution (u, v, w, z) will converge to r/μ1α−1,v̄0+w̄0,0,r/μ1α−1 as t → ∞.

Global solvability and asymptotic behavior of solutions for a fully parabolic nutrient taxis system

In this paper, we consider the fully parabolic nu’trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights.

Global asymptotical stability and Hopf bifurcation for a three‐species Lotka‐Volterra food web model

In this article, we consider a delayed three‐species Lotka‐Volterra food web model with diffusion and homogeneous Neumann boundary conditions. We proved that the positive constant equilibrium solution is globally asymptotically stable for the system without time delays. By virtue of the sum of time delays as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant equilibrium solution are proved to occur when the delay varied through a sequence of critical values. In addition, we consider the effect of cross‐diffusion on the system in the case that without time delays. By taking cross diffusion coefficients as the bifurcation parameter, our model undergoes inhomogeneous Hopf bifurcation around a positive constant equilibrium solution when the bifurcation parameter is varied through a sequence of critical values. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms (such as time delay) rather than diffusion terms. Our results demonstrate that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross diffusion factors.

Comparing coherent and incoherent models for quantum homogenization

Here we investigate the role of quantum interference in the quantum homogenizer, whose convergence properties model a thermalization process. In the original quantum homogenizer protocol, a system qubit converges to the state of identical reservoir qubits through partial-swap interactions, that allow interference between reservoir qubits. We design an alternative, incoherent quantum homogenizer, where each system-reservoir interaction is moderated by a control qubit using a controlled-swap interaction. We show that our incoherent homogenizer satisfies the essential conditions for homogenization, being able to transform a qubit from any state to any other state to arbitrary accuracy, with negligible impact on the reservoir qubits' states. Our results show that the convergence properties of homogenization machines that are important for modeling thermalization are not dependent on coherence between qubits in the homogenization protocol. We then derive bounds on the resources required to re-use the homogenizers for performing state transformations. This demonstrates that both homogenizers are universal for any number of homogenizations, for an increased resource cost. Published by the American Physical Society 2024

Global dynamics and spatiotemporal patterns of a two‐species chemotaxis system with chemical signaling loop and Lotka–Volterra competition

AbstractThis paper considers a two‐species chemotaxis system with chemical signaling loop and Lotka–Volterra competition kinetics under the homogeneous Newman boundary condition in smooth bounded domains. The global existence and boundedness of solutions for the parabolic–elliptic/parabolic–parabolic system are established. In the strong competition case, the global stability of the semitrivial constant steady state is obtained under certain parameter conditions. Linear analyzes and numerical simulations demonstrate that chemical signaling loop can significantly impact population dynamics, and admit the coexistence in the exclusion competitive case, including nonconstant steady states, chaos, and spatially inhomogeneous time‐periodic types.

Superlinear transmission in an indirect signal production chemotaxis system

In this paper, the indirect signal production system with nonlinear transmission is considered ut=Δu−∇⋅(u∇v),vt=Δv−v+w,wt=Δw−w+f(u)in a bounded smooth domain Ω⊂Rn (n≥1) associated with homogeneous Neumann boundary conditions, where f∈C1([0,∞)) satisfies 0≤f(s)≤sα with α>0. It is known from [1] that the system possesses a global bounded solution if 0<α<4n when n≥4. In the case n≤3 and if we consider superlinear transmission, no regularity of w or v can be derived directly. In this work, we show that if 0<α<min{4n,1+2n}, the solution is global and bounded via an approach based on the maximal Sobolev regularity.

Elliptic equations in weak oscillatory thin domains: Beyond periodicity with boundary‐concentrated reaction terms

In this paper, we analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two‐dimensional thin domain that presents weak oscillations on both boundaries and with terms concentrated in a narrow oscillating neighborhood of the top boundary. The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions,including scenarios such as quasi‐periodic or almost‐periodic oscillations. We then prove that the family of solutions converges to the solution of a one‐dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition, we include a series of numerical experiments illustrating the theoretical results obtained in the quasi‐periodic context.

Heterogeneity promotes resilience in restored prairie: Implications for the environmental heterogeneity hypothesis.

Enhancing resilience in formerly degraded ecosystems is an important goal of restoration ecology. However, evidence for the recovery of resilience and its underlying mechanisms require long-term experiments and comparison with reference ecosystems. We used data from an experimental prairie restoration that featured long-term soil heterogeneity manipulations and data from two long-term experiments located in a comparable remnant (reference) prairie to (1) quantify the recovery of ecosystem functioning (i.e., productivity) relative to remnant prairie, (2) compare the resilience of restored and remnant prairies to a natural drought, and (3) test whether soil heterogeneity enhances resilience of restored prairie. We compared sensitivity and legacy effects between prairie types (remnant and restored) and among four prairie sites that included two remnant prairie sites and prairie restored under homogeneous and heterogeneous soil conditions. We measured sensitivity and resilience as the proportional change in aboveground net primary productivity (ANPP) during and following drought (sensitivity and legacy effects, respectively) relative to average ANPP based on 4 pre-drought years (2014-2017). In nondrought years, total ANPP was similar between remnant and restored prairie, but remnant prairie had higher grass productivity and lower forb productivity compared with restored prairie. These ANPP patterns generally persisted during drought. The sensitivity of total ANPP to drought was similar between restored and remnant prairie, but grasses in the restored prairie were more sensitive to drought. Post-drought legacy effects were more positive in the restored prairie, and we attributed this to the more positive and less variable legacy response of forb ANPP in the restored prairie, especially in the heterogeneous soil treatment. Our results suggest that productivity recovers in restored prairie and exhibits similar sensitivity to drought as in remnant prairie. Furthermore, creating heterogeneity promotes forb productivity and enhances restored prairie resilience to drought.

Asymptotic Behavior and Numerical Simulations of a Conservative Phase-Field Model with Two Temperatures

One of the types of problem that has attracted the attention of mathematicians in recent years is the phase field system. The field of application includes materials science, where phenomena such as phase separation in alloys, crystal formation and thermal welding are legion. Among these phase transition systems, the family of conservative systems is very popular with industry. Indeed, minimising losses in production systems is a major issue for their profitability. In this paper, we study the well-posedness of the formulation and the asymptotic behaviour of the solutions, by proving the existence of a finite-dimensional global attractor for a conservative variant of the two-temperature phase field system with homogeneous Neumann boundary conditions. The inclusion of two temperatures in the definition of the enthalpy of the system is a necessity in the case of a non-simple material. In a simple material, once the phase-change temperature has been reached, the temperature of the system remains constant until the material has completely changed state. This is not true in the case of a non-simple material, where an increase in the temperature of the system is observed even after the phase-change temperature has been reached. To conclude the work, we present a method for numerically approximating the solution and carry out some numerical tests.

Patchy energy landscapes promote stability of small groups of active particles

Group formation and coordination are fundamental characteristics of living matter, essential for performing tasks and ensuring survival. Interactions between individuals play a key role in group formation, and the impact of resource distributions is a vibrant area of research. As of now, an understanding of how patchy resource distributions determine group dynamics is not yet fully understood. Studying active particles in controlled optical landscapes as energy sources, we demonstrate a non-monotonic dependency of group size on landscape patchiness, with the smallest groups forming when the patches match the active particles’ size. A similar relationship is observed in terms of group stability, evidenced by a reduced rate of individual exchange in patchy environments compared to homogeneous conditions. Reduced group sizes can be beneficial to optimise resources in heterogeneous environments and to control information flow within populations. Our results provide insights into the role of patchy landscapes and uneven energy distributions in active matter and hold implications for refining swarm intelligence algorithms, enhancing crowd management techniques, and tailoring colloidal self-assembly.

Spatio-temporal dynamics in a delayed prey–predator model with nonlinear prey refuge and harvesting

In the current study, we introduce a model for the temporal and spatial interactions between prey and predator. The model incorporates a nonlinear refuge mechanism for prey, along with linear harvesting of prey and nonlinear harvesting for predators. Initially, we examined the well-posed nature of the model by analyzing the presence of all feasible equilibria and investigating the corresponding dynamics. Following that, we delve into the dynamics of the temporal model, focusing specifically on aspects such as uniform boundedness, permanence, and stability of viable equilibria. We demonstrate analytically that the proposed model experiences transcritical, saddle–node, Hopf, and Bogdanov–Takens bifurcations. It shows a variety of intricate dynamics involving Generalized Hopf and double Hopf. Then, discrete-time delay effects arising from the gestation of predator species have been incorporated into the temporal system. Hopf bifurcation for the delay parameter was detected in this investigation. Subsequently, we established conditions for self-diffusion instability and Turing instability in the spatiotemporal model, both with and without delay, employing the homogeneous Neumann boundary condition. Moreover, the discussion of sensitivity analysis (PRCC) serves to illustrate how crucial parameters influence the dynamics of the system. In addition, we conduct numerical simulations aiming to corroborate and validate the analytical results obtained.

Toward Ultra-High-Quality-Factor Wireless Masing Magnetic Resonance Sensing.

It has recently been shown that a bolus of hyperpolarized nuclear spins can yield stimulated emission signals similar in nature to maser signals, potentially enabling new ways of sensing hyperpolarized contrast media, including most notably [1-13C]pyruvate that is under evaluation in over 50 clinical trials for metabolic imaging of cancer. The stimulated NMR signal emissions lasting for minutes do not require radio-frequency excitation, offering unprecedented advantages compared to conventional MR sensing. However, creating nuclear spin maser emission is challenging in practice due to stringent fundamental requirements, making practical in vivo applications hardly possible using conventional passive MR detectors. Here, we demonstrate the utility of a wireless NMR maser detector, the quality factor of which was enhanced 22-fold (to 1,670) via parametric pumping. This active-feedback technique breaks the intrinsic fundamental limit of NMR detector circuit quality factor. We show the use of parametric pumping to reduce the threshold requirement for inducing nuclear spin masing at 300 MHz resonance frequency in a preclinical MRI scanner. Indeed, stimulated emission from hyperpolarized protons was obtained under highly unfavorable conditions of low magnetic field homogeneity (T2* of 3 ms). Greater gains of the quality factor of the MR detector (up to 1 million) were also demonstrated.

Toward Ultra‐High‐Quality‐Factor Wireless Masing Magnetic Resonance Sensing

AbstractIt has recently been shown that a bolus of hyperpolarized nuclear spins can yield stimulated emission signals similar in nature to maser signals, potentially enabling new ways of sensing hyperpolarized contrast media, including most notably [1‐13C]pyruvate that is under evaluation in over 50 clinical trials for metabolic imaging of cancer. The stimulated NMR signal emissions lasting for minutes do not require radio‐frequency excitation, offering unprecedented advantages compared to conventional MR sensing. However, creating nuclear spin maser emission is challenging in practice due to stringent fundamental requirements, making practical in vivo applications hardly possible using conventional passive MR detectors. Here, we demonstrate the utility of a wireless NMR maser detector, the quality factor of which was enhanced 22‐fold (to 1,670) via parametric pumping. This active‐feedback technique breaks the intrinsic fundamental limit of NMR detector circuit quality factor. We show the use of parametric pumping to reduce the threshold requirement for inducing nuclear spin masing at 300 MHz resonance frequency in a preclinical MRI scanner. Indeed, stimulated emission from hyperpolarized protons was obtained under highly unfavorable conditions of low magnetic field homogeneity (T2* of 3 ms). Greater gains of the quality factor of the MR detector (up to 1 million) were also demonstrated.

Local exponential stabilisation of the delayed Fisher's equation

Exponential stabilisation is undoubtedly an essential property to study for any control system. This property is often difficult to establish for nonlinear distributed systems and even more difficult if the state of such a system is subject to a time-delay. In this context, this paper is devoted to the study of the well-posedness and exponential stabilisation problems for the one-dimensional delayed Fisher's partial differential equation (PDE) posed in a bounded interval, with homogeneous Dirichlet condition at the left boundary and controlled at the right boundary by Dirichlet condition. Based on the infinite dimensional backstepping method for the linear and undelayed corresponding case, a feedback law is built. Under this feedback, we prove that the closed-loop system is well-posed in L 2 ( 0 , 1 ) and locally exponentially stable in the topology of the L 2 ( 0 , 1 ) -norm by the use of semigroup theory relating to differential difference equations and Lyapunov–Razumikhin analysis, respectively. A numerical example that reflects the effectiveness of the proposed approach is given.

Global dynamics of a diffusive SIRS epidemic model in a spatially heterogeneous environment

In this paper, we have proposed a reaction-diffusion SIRS epidemic model with a general incidence function f ( S , I ) in a spatially heterogeneous environment. For this model, we derive the basic reproduction number R 0 and establish the results of the threshold dynamics with respect to the basic reproduction number R 0 . Specifically, the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , while the disease persists if 1 $ ]]> R 0 > 1 . Especially, under the spatially homogeneous condition, the SIRS model admits a unique steady state, which is globally asymptotically stable under some assumptions when 1 $ ]]> R 0 > 1 . Finally, we take the Beddington-DeAngelis-type incidence function and perform some numerical simulations to illustrate the dynamics of the solutions as the model parameters are varied.

Optimal Neumann control of the wave equation with L 1-control cost: the finite-time turnpike property

The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L 1 -norm control cost with a weight γ > 0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L 2 -norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L 1 -norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L 2 -sense.

Emergence of boundary conditions in the heat equation

The Dirichlet and Neumann conditions are commonly employed as boundary conditions for the heat equation, yet their legitimacy is debatable in certain scenarios. This paper aims to demonstrate that, in fact, diffusion laws autonomously select boundary conditions. To illustrate this, we incorporate the bounded domain into a larger domain with a diffusivity parameter ϵ &amp;gt; 0 and examine the solution’s behavior at the interface. Our findings reveal that homogeneous Neumann or Dirichlet boundary conditions emerge as ϵ → 0, contingent upon the type of the heterogeneous diffusion.

The influence of hillslope topography on beech water use: a comparative study in two different climates

Abstract. Understanding the interrelation between topography and vegetation across different environments is important to assess how hydrological and climatic conditions affect tree physiological activity. This becomes especially important given the expected reduction in water availability and the increase in water demand driven by climate change. These extremes could enhance the thermal and hydrologic gradients along slopes. Here, we aimed to test if and how different climatic and hydrological conditions affect the physiological response of beech trees (Fagus sylvatica L.) to environmental variables along two different topographic sequences. For this purpose, we set up a comparative study on a gentle hillslope in the Weierbach catchment in Luxembourg (oceanic climate) and on a steep hillslope in the Lecciona catchment in Italy (Mediterranean climate). We combined sap velocity measurements with isotopic measurements of soil, precipitation, stream water, groundwater, and xylem over 2019 and 2020 for the Luxembourgish site and over 2021 for the Italian site. We found that, in the Weierbach catchment, trees' responses to environmental variables (i.e. vapour pressure deficit and relative extractable water in the soil) were similar among hillslope positions and between the two monitored years, resulting from homogeneous growing conditions along the topographic sequence. We also did not find any statistical difference in the isotopic composition of xylem water between positions, suggesting that beech trees relied on similar water sources across the landscape. In the Lecciona catchment, we observed lower sap velocities and shorter growing season in trees growing in the upper portions of the hillslope, likely related to water redistribution and different soil moisture along the hillslope catena. Xylem isotopic composition was significantly lighter at the footslope location throughout the growing season than in the upslope locations, suggesting location-specific water use. These results emphasize how differing hydrometeorological processes occurring at the hillslope scale can lead to contrasting tree responses.