We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables for such solutions into a wide class of CAT(0) spaces, answering a long-standing open problem in the field. Our approach is based on an elliptic regularization of the gradient flow of the Dirichlet energy and even in the case of smooth Riemannian targets provides a novel viewpoint, together with a new Dynamical Variational Principle and a new proof of the celebrated Eells-Sampson theorem. The spatial Lipschitz regularity for such weak solutions is achieved by fully exploiting the variational structure of the problem at the regularized level and introducing a parabolic frequency function of Almgren-Poon type. Our contribution is the first instance of the use of monotonicity methods for parabolic deformations of maps into singular targets.

In this paper, we present an alternate, elementary proof of the local Lipschitz regularity of the suitable weak solution of heat flow of harmonic maps into CAT(0)-metric spaces, whose existence was established by Lin, Segatti, Sire, and Wang through an elliptic regularization approach. The ideas of the proof are inspired by Korevaar and Schoen, and they work for any CAT(0)-metric space (X,d) as the target and any complete Riemanan manifold (M,g), with positive injectivity radius and bounded curvature, as the domain.

This paper investigates a class of predator-prey systems with time delays, stochastic perturbations, and patch diffusion, which incorporate the Beddington–DeAngelis (B-D) functional response. The proposed model integrates environmental noise, dispersal processes, and time delays, forming a comprehensive stochastic delayed diffusion framework that better reflects realistic ecological scenarios. By employing appropriate Lyapunov functions and the comparative principle of stochastic differential equations, we prove the existence of a unique global positive solution for any given positive initial value. Sufficient conditions are also established for uniform persistence of the system. Furthermore, through the construction of a novel Lyapunov functional, we derive sufficient conditions for the global attractivity of the system’s solutions, providing theoretical insights into the long-term dynamical behavior under the combined effects of stochasticity, delays, and diffusion. Finally, numerical simulations are conducted using MATLAB along with the Milstein discretization method, which corroborate the correctness and effectiveness of the theoretical results. This study offers a new theoretical framework and methodology for analyzing persistence and stability in stochastic delayed diffusive predator-prey systems.

Max-type fuzzy difference equations constitute an emerging research area arising from the integration of fuzzy mathematics and discrete dynamical systems. By characterizing uncertainty through fuzzy numbers, these equations provide rigorous mathematical modeling tools for practical problems involving both discreteness and uncertainty. This paper systematically investigates the dynamical properties of a class of max-type fuzzy difference equations. First, fuzzy set theory is used to transform the fuzzy difference equation into a corresponding system of parametric ordinary difference equations. Then, using the iterative method, inequality techniques, and mathematical induction, an expression for the periodic solutions of the max-type ordinary difference equation is derived, from which an expression for the periodic solutions of the max-type fuzzy difference equation is further obtained. In addition, the boundedness and persistence of the solutions to the fuzzy difference equation are proved. Finally, numerical simulations are conducted in MATLAB 2024, and the results illustrate the theoretical findings. This study not only enriches the theoretical framework of fuzzy difference equations but also provides new insights and methodological support for the modeling and analysis of uncertain discrete systems.

In this paper, we establish an angular energy quantization for the following fourth order inhomogeneous Lamm-Rivière system Δ^2u=Δ(V·∇ u)+div(w∇ u)+W·∇ u+f in dimension four, with an inhomogeneous term f∈ Llog L.

The purpose of this note is to present a positive answer to an open problem proposed in the recent book by H. Brezis and P. Mironescu. It has been stated in this book Sobolev Maps to the Circle as Proposition 4.3. We demonstrate, in particular, the value of the least mass of the area minimizing integral rectifiable currents with a given boundary equals to the infimum of areas among smoothly immersed submanifolds with the same boundary, under the assumption that the boundary is that of a smooth submanifold.

Energy identity for harmonic type maps in supercritical dimensions is an important and difficult problem. For sphere-valued harmonic maps, the first breakthrough was achieved by Lin-Rivière [Duke Math. J. 2002]. In this paper, by adapting their strategy, we establish the energy identity for stationary biharmonic maps into spheres in supercritical dimensions n≥ 5.

This paper systematically investigates the dynamical properties of a class of max-type fuzzy difference equation. The study first establishes the existence and uniqueness of the solution sequence under given initial conditions with positive fuzzy numbers. Subsequently, by applying the cut-set theory, the fuzzy equation is transformed into a system coupled by two ordinary difference equations. Through a combination of case analysis and mathematical induction, the study rigorously demonstrates that the solutions of this system exhibit global periodicity with a period of 4, while also deriving the exact closed-form expressions of the periodic solutions. Based on the periodic solutions obtained from the ordinary difference system, the research successfully reveals the periodic characteristics of the solutions to the original fuzzy difference equation and rigorously analyzes their boundedness and persistence. Finally, numerical simulations conducted with Matlab 2016 provide robust data support for the theoretical conclusions and the effectiveness of the methodology.

n this paper, we revisit the existence of global weak solutions of wave maps from ^n into the sphere 𝕊^L-1, u⊥ T_u 𝕊^L-1, by establishing it as a singular limit of maps from ^n×_+ to 𝕊^L-1 that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter ε, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi , which has been implemented by Serra and Tilli for certain class of nonlinear wave equations. This approach is also applicable to the SO(m)-target manifold.

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of the celebrated harmonic extension method by Caffarelli-Silverstre. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.

Noctiluca scintillans is one of the most common harmful algal species worldwide. In this study, a MaxEnt model was constructed to calculate the present and future habitat suitability of N. scintillans in the China Sea. A comprehensive evaluation index of variable importance was defined to measure the importance of key predictors in the model, and offshore distance, long-term average minimum primary productivity, water depth, long-term average minimum temperature, and minimum salinity were determined as the dominant drivers. The HAB index that was constructed by integrating the carrying capacity and habitat suitability characteristics was used to measure the risk of harmful algal blooms (HABs). The index indicated that high-risk areas of HABs caused by N. scintillans occurred around Hainan Island, Taiwan Island, the coastal areas of Guangdong, Fujian, and Zhejiang provinces, and the eastern nearshore area of Weihai in Shandong. Regardless of the greenhouse gas emission scenario, the occurrence of N. scintillans red tides was predicted to persist until 2100. Moreover, the total area of the HABs high zone was predicted to increase under RCP2.6 and decrease under RCP8.5, and the center of the integrated HABs high zone was predicted to be concentrated in the central area of the entire China Sea spanning 15°N to 33°N.

In this study, we research a nonautonomous, three-species, delayed reaction–diffusion predator–prey model (RDPPM). Firstly, we derive sufficient conditions to guarantee the existence of a strictly positive, spatially homogeneous periodic solution (SHPS) for the delayed, nonautonomous RDPPM. These conditions are obtained using the comparison theorem for delayed differential equations and the fixed point theorem. Secondly, we present sufficient conditions to ensure the global asymptotic stability of the SHPS for the delayed, nonautonomous RDPPM. These conditions are established through the application of the upper and lower solution method (UALSM) for delayed parabolic partial differential equations (PDEs), along with Lyapunov stability theory. Finally, to demonstrate the practical application of our results, we numerically validate the proposed conditions using a 2-periodic, delayed, nonautonomous RDPPM.

In this article, the well-posedness of a generalized discrete-time Riccati model with a fuzzy environment is studied. Initially, relying on the theory of fuzzy sets and the arithmetic operations for fuzzy numbers, the generalized discrete-time Riccati model with a fuzzy environment is reformulated into the common difference equations, incorporating parameters with a value scope confined between 0 and 1. Secondly, the proof of the existence and uniqueness of solutions for the common difference equations is accomplished using a blend of iterative strategies, inductive proofs, and inequality manipulations. At the same time, the stability of the equilibrium solution of p + q + r = 1 in a special case is discussed according to the linearized equation theory. Thus, the well-posedness of the generalized discrete-time Riccati model with a fuzzy environment is proven.

For a ball BR(0) C Ilg2, we provide sufficient conditions such that a harmonic map u E C infinity(BR(0) \{0}, N), with a self-similar bound on its gradient, belong to C infinity(BR(0)). These conditions also guarantee the triviality of such harmonic maps when R = oo.

Based on existing models, this paper incorporates some key ecological factors, thereby obtaining a class of eco-epidemiological models that can more objectively reflect natural phenomena. This model simultaneously integrates disease dynamics within the prey population and the Beddington-DeAngelis functional response, thus achieving an organic combination of ecological dynamics, epidemic transmission, and spatial movement under time-varying environmental conditions. The proposed framework significantly enhances ecological realism by simultaneously accounting for spatial dispersal, predator-prey interactions, disease transmission within prey species, and seasonal or temporal variations, providing a comprehensive mathematical tool for analyzing complex eco-epidemiological systems. The theoretical results obtained from this study can be summarized as follows: Firstly, the existence and uniqueness of globally positive solutions for any positive initial data are rigorously established, ensuring the well-posedness and biological feasibility of the model over extended temporal scales. Secondly, analytically tractable sufficient conditions for uniform population persistence are derived, which elucidate the mechanisms of species coexistence and biodiversity preservation even under sustained epidemiological pressure. Thirdly, by employing innovative applications of differential inequalities and fixed point theory, the existence and uniqueness of a positive spatially homogeneous periodic solution in the presence of time-periodic coefficients are conclusively demonstrated, capturing essential rhythmicities inherent in natural systems. Fourthly, through a sophisticated combination of the upper and lower solution method for parabolic partial differential equations and Lyapunov stability theory, the global asymptotic stability of this periodic solution is rigorously established, offering a powerful analytical guarantee for long-term predictive modeling. Beyond theoretical contributions, these research findings provide actionable insights and quantitative analytical tools to tackle pressing ecological and public health challenges. They facilitate the prediction of thresholds for maintaining ecosystem stability using real-world data, enable the analysis and assessment of disease persistence in spatially structured environments, and offer robust theoretical support for the planning and design of wildlife management and conservation strategies. The derived criteria support evidence-based decision-making in areas such as controlling zoonotic disease outbreaks, maintaining ecosystem stability, and mitigating anthropogenic impacts on ecological communities. A representative numerical case study has been integrated into the analysis to verify all of the theoretical findings. In doing so, it effectively highlights the model's substantial theoretical value in informing policy-making and advancing sustainable ecosystem management practices.

This article focuses on the stability of periodic solution of a delayed nonautonomous reaction-diffusion predator-prey model. The fine combination of upper and lower solution methods and Lyapunov stability theory is used to transform the study of the stability problem of delayed reaction- diffusion equations into the stability problem of their corresponding delayed ordinary differential equations. Some sufficient conditions are given to ensure the globally asymptotically stability of the periodic solution for this model. Unlike existing results, the stable solution which are obtained in this article is a time-periodic solution rather than a constant periodic solution or a solution for a steady-state system. We extend a stability theorem on predator-prey model introduced by V. Ortega and C. Rebelo in 2023 to nonautonomous delayed reaction-diffusion model. Finally, in order to show the application of the theoretical results, the proposed conditions are numerically validated over a 2-periodic delayed nonautonomous reaction-diffusion predator-prey model.

This study develops a novel non-autonomous diffusion-driven food chain model incorporating time-varying delays, feedback control mechanisms, and a Michaelis-Menten functional response to address limitations in traditional ecological models. By integrating spatiotemporal dynamics with realistic biological interactions, we derive rigorous theoretical conditions for the existence and global asymptotic stability of spatially homogeneous periodic solutions using a hybrid analytical framework combining fixed point theory, Lyapunov functionals, and limit approximation methods. The model accounts for environmental fluctuations and delayed responses, revealing critical dependencies of population persistence on diffusion rates, feedback control strengths, and delay structures. Numerical simulations parameterized with empirical data validate these findings, demonstrating delay-induced oscillations, trophic cascade propagation, and feedback-mediated stabilization under environmental stochasticity. Results highlight the model’s capacity to predict population resilience in changing environments, offering mechanistic insights for conservation biology, pest management, and ecosystem resilience assessment. This work bridges a gap between theoretical reaction-diffusion systems and complex ecological realities, providing a foundational framework for predicting species responses to climate change and habitat fragmentation while emphasizing the stabilizing role of adaptive feedback in maintaining ecosystem balance.

Neural networks, mimicking the structural and functional aspects of the human brain, have found widespread applications in diverse fields such as pattern recognition, control systems, and information processing. A critical phenomenon in these systems is synchronization, where multiple neurons or neural networks harmonize their dynamic behaviors to a common rhythm, contributing significantly to their efficient operation. However, the inherent complexity and nonlinearity of neural networks pose significant challenges in understanding and controlling this synchronization process. In this paper, we focus on the synchronization of a class of fractional-order, delayed, and non-autonomous neural networks. Fractional-order dynamics, characterized by their ability to capture memory effects and non-local interactions, introduce additional layers of complexity to the synchronization problem. Time delays, which are ubiquitous in real-world systems, further complicate the analysis by introducing temporal asynchrony among the neurons. To address these challenges, we propose a straightforward yet powerful global synchronization framework. Our approach leverages novel state feedback control to derive an analytical formula for the synchronization controller. This controller is designed to adjust the states of the neural networks in such a way that they converge to a common trajectory, achieving synchronization. To establish the asymptotic stability of the error system, which measures the deviation between the states of the neural networks, we construct a Lyapunov function. This function provides a scalar measure of the system’s energy, and by showing that this measure decreases over time, we demonstrate the stability of the synchronized state. Our analysis yields sufficient conditions that guarantee global synchronization in fractional-order neural networks with time delays and Caputo derivatives. These conditions provide a clear roadmap for designing neural networks that exhibit robust and stable synchronization properties. To validate our theoretical findings, we present numerical simulations that demonstrate the effectiveness of our proposed approach. The simulations show that, under the derived conditions, the neural networks successfully synchronize, confirming the practical applicability of our framework.

Harmful dinoflagellates are widely distributed in coastal waters worldwide, posing multiple ecological and socioeconomic threats. Climate change may alter the biogeography of these species; however, few studies have linked shifts in harmful dinoflagellates' ecological distribution to their socioeconomic impacts. This study developed a framework to assess the spatiotemporal ecological-social risks posed by harmful dinoflagellates, identifying these algae as risk sources and considering mariculture and coastal populations as the primary risk receptors. China is the world's largest mariculture producer, with approximately 600 million residents living in coastal areas. Focusing on 14 key harmful dinoflagellate species in Chinese coastal waters, we evaluated ecological-social risks under present conditions and two projected climate scenarios for 2100. Our findings indicate that climate change may lead to reductions in suitable habitats for harmful dinoflagellates in tropical and subtropical regions, while habitats in higher-latitude areas are likely to remain stable or expand. Risk area expansion is projected for four species and increased average risk intensity for three, with two species experiencing both. Nationally, total risk area is projected to remain stable, while cumulative risk intensity may decline by 16.64%. Regionally, risk intensity is expected to rise in northern provinces (up to 30.46%) and decline across most southern provinces. Importantly, we reveal a potential spatial "decoupling" of risk sources and receptors along the coast of China in the future. This decoupling demonstrates a reduced overlap between harmful dinoflagellate distributions and areas with dense mariculture or populations. Our findings suggest that, contrary to the common assumption that climate change universally exacerbates harmful algal impacts, these effects may vary across regions and species, highlighting the importance of localized adaptation strategies in risk assessment. This study provides a robust tool for understanding harmful dinoflagellate risks under climate change, thereby supporting the sustainable management of coastal ecosystems.

For any k-dimensional smooth, compact Riemannian manifold (N, h)⊂ℝ^L without boundary, there exists an ε _0>0 such that for any homogeneous of degree zero map u_0(x)=ϕ _0(x/|x|):ℝ^n→ N ( n≥ 2 ), if ‖∇ϕ _0‖ _L^n(𝕊^n-1)≤ε _0 then there is a unique solution u:ℝ^n× (0,∞ )→ N to the heat flow of harmonic map (1.1) and (1.2), which is forward self-similar and belongs to C^∞ (ℝ^n× (0,∞ ))∩ C^1/n(ℝ^n× [0,∞ )∖{(0,0)})