This paper is concerned with the investigation of the effect of a boundary infinite memory term on the stability of the nonlinear rotating disk-beam system. Assuming that infinite memory is of angular velocity type, the minimal state approach is employed to handle the memory term. Under specific conditions on the memory kernel function and the physical parameters of the system, we demonstrate that the problem is well-posed and its solutions are exponentially stable. In particular, the beam vibrations are suppressed, and the disk keeps rotating at a desired angular velocity, provided the latter remains bounded. Last but not least, using the Finite Volumes Method, a comprehensive numerical study validates the theoretical stability results.

The main results of this paper are the global existence and long time behavior of solutions of a fractional wave equation with a nonlocal nonlinearity. The techniques in this work rely on norm estimates of the solutions of ε u tt + u t + ( - Δ ) β u = 0 , u ( 0 , x ) = φ ( x ) , u t ( 0 , x ) = ψ ( x ) , which we derive particularly to observe the roles of ε and β in the long time behavior of solutions. Moreover, we apply these estimates to obtain local in time weak solutions, and global solutions, under the influence of a nonlocal non-power nonlinear term.

This paper is concerned with the large-time behavior of solutions to the one-dimensional full compressible Navier-Stokes-Allen-Cahn system with a free boundary. The model can be used to describe the motion of a mixture of two viscous compressible fluids. We first construct a viscous contact wave which approximates to the contact discontinuity, which is a basic wave pattern of compressible Euler equation. Then we prove that the viscous contact wave is time-asymptotically stable, provided that the strength of contact wave and the initial perturbation are sufficiently small. The proof is given by the elementary L 2 -energy method.

We present a computationally simple yet efficient method for planning global trajectories in deterministic dynamic environments of arbitrary dimensions and complexity. The method leverages morphological operations on a discrete Minkowski spacetime and constructs cognitive maps that encode the earliest arrival time at each point. We established a solid theoretical foundation, ensuring the approach's applicability in risk-critical missions. The method is illustrated using simulations of terrestrial robots and flying drones, resulting in efficient yet conservative solutions that closely mimic the behavior of biological agents.

In this paper, f--Dirac--harmonic maps are studied. then, we monotonicity formulas for solutions of these equations are derived. Also, Liouvile theorem for f--Dirac--harmonic maps is proved.

In this paper, the spatiotemporal pattern dynamics of a diffusion population model with Allee effect and environmental toxins is considered. Firstly, the boundedness and positivity of the system are proved, the existence and stability of the equilibrium are analyzed, and the conditions of the occurrence of Turing bifurcation are deduced. Subsequently, the mechanism for the selection of patterns around the positive equilibrium was provided through weakly nonlinear analysis. One finds that weak Allee effect can restore the reduction of population density and undergoes the bistable phenomena. Environmental toxins can lead to Hopf bifurcation and saddle-node bifurcation, and change the topology of the system. Self-diffusion can induce Turing bifurcation, and the hexagonal hole patterns, mixed patterns of holes and stripes, and striped patterns are shown. Finally, theoretical results are validated through numerical simulations. These findings provide a scientific basis for population conservation strategies under the threat of environmental toxins.

This study introduces a novel fractional thermoelastic model to analyze the behavior of isotropic three-dimensional solids by combining the Green-Naghdi framework, the two-temperature theory, and the Moore-Gibson-Thompson (MGT) equation. The approach incorporates nonlocal memory effects through a two-parameter Mittag-Leffler fractional derivative, offering a more sophisticated representation of thermal interactions. The model specifically investigates the response of materials under free surface conditions exposed to laser-induced convection, characterized by a Gaussian distribution in both time and space. Analytical solutions are derived using a combination of Laplace and multi-Fourier transform techniques, supplemented by numerical simulations to assess the influence of key parameters. The results reveal that the two-temperature parameter significantly affects the amplitude and phase of thermal and mechanical responses, while the relaxation time and fractional-order coefficients determine the decay rates and oscillation patterns of system behavior. This fractional thermoelastic perspective provides deeper insights into thermal dynamics, paving the way for advanced engineering applications in thermal insulation design, high-efficiency heat exchangers, and next-generation energy storage systems.

A new approximation function is introduced to fit the solution of a fractional differential equation of order one-half. The analyzed case includes a nonhomogeneous term defined by a modified Bessel function of the first kind. The analytical solution of this equation corresponds to the product of two modified Bessel functions. The new fitting function is developed using the multipoint quasi-rational method, which uses both the series expansion of the Bessel function and its asymptotic expansion. Additionally, a significant modification is introduced to the structure of the fitting function to capture two terms of the asymptotic expansion of the Bessel functions. We show an example for fixed values, and the maximum relative error of the fitting function for the solution of the fractional differential equation of order one-half is 0 .18%, a remarkably low value considering that only six fitting parameters are used.

The Turing pattern of a population model with cross-diffusion and prey defense are studied in this paper. According to the existence conditions of Turing bifurcation, we derive the amplitude equation of the Turing bifurcation of the model at its positive equilibrium by the multi-scale analysis method. By analyzing the topologically equivalent amplitude equation, one find that cross-diffusion can induce Turing instability in the population system at the positive equilibrium, resulting in rich pattern dynamics phenomena, such as stripes, spots or hexagonal patterns. And the change of the prey defense coefficient will trigger the backward Hopf bifurcation, indicating that the prey defense mechanism is beneficial for maintaining the stability of the population system. Finally, the theoretical results are proved numerically.

This paper investigates the dynamics of a slow-fast Leslie-Gower predator-prey model incorporating a generalized Holling type II functional response, prey harvesting, and a weak Allee effect in the predator population. Combining the normal form theory of slow-fast systems and the geometric singular perturbation theory, we address the rich canard phenomena including the existence of canard cycles, singular Hopf bifurcation, homoclinic and heteroclinic orbits, birth of canard explosion. We also employ the entry-exit function to demonstrate the presence of relaxation oscillations. Furthermore, we observe much richer new dynamical phenomena, specifically examining the transformation of bistability via Hopf bifurcation. Our results demonstrate that variations in parameters and initial population sizes can lead to different long-term outcomes, ranging from predator extinction to stable coexistence. The theoretical results are confirmed by numerical simulations.

We study the asymptotic behavior of trajectory attractors of the Ginzburg–Landau complex equation in a perforated domain with a rapidly oscillating outer boundary when the parameter characterizing the pore sizes and the distance between them, as well as the amplitude and frequency of the boundary oscillation, tends to zero. The asymptotic behaviour of attractors to an initial boundary value problem for complex Ginzburg–Landau equations in perforated domains for the critical case (appearance of additional potential in the homogenized equation) is studied by Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., and Tolemis, A.A. in “Homogenization of Attractors to Ginzburg–Landau Equations in Media with Locally Periodic Obstacles: Critical Case” (Bulletin of the Karaganda university. Mathematics series.- 2023.- v. 3 (111).- p. 11–27). By defining the auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main Theorem of weak convergence of attractors and prove it on the base of auxiliary lemmas.

This paper investigates the stability of perturbations near equilibrium states for the two-dimensional Boussinesq-magnetohydrodynamics(MHD) system with fractional horizontal dissipation in the domain Ω=T×R, where T=[0 ,1] is the one-dimensional periodic box. The absence of vertical dissipation, combined with the presence of fractional operators, presents significant challenges to the study of stability and long-time dynamics in the Boussinesq-MHD system. To address these issues, we leverage the orthogonal decomposition method in conjunction with multiple fractional anisotropic interpolation inequalities and a fractional Poincaré-type inequality within our analytical framework. We prove the global stability of the system when the initial data are sufficiently small in H 2 ( Ω ) . Moreover, as time tends to infinity, the oscillatory part of the solution converges exponentially to zero in H 1 ( Ω ) .

Inspired by the growth dynamics of the protist Physarum polycephalum, we apply the formalism describing adaptive incompressible Hagen-Poiseuilly flows on channel networks to find graphs connecting different nodes distributed on an Euclidean space. These graphs are suboptimal or optimal relative to the graph length. We sometimes obtain graph tree configurations topologically equivalent to Steiner trees. This approach is applied to design communication systems, such as railway systems, highways, or fibre webs. As a proof of concept, we explicitly apply this framework to the Portuguese railway systems.

In this paper, we are concerned with global existence and temporal decay rates of large solutions for a mathematical model arising from electro-hydrodynamics, which is the nonlinear dissipative system coupled by the Poisson-Nernst-Planck equations and the incompressible Navier-Stokes equations through charge transport and external forcing terms. By introducing some proper weighted functions based on carefully examining the algebraic structure of the system, we prove that there exist two positive constants c 0 , C 0 such that if the initial data ( u 0 , N 0 , P 0 ) satisfies ( ∥ u L · ∇ u L ∥ L 1 ( B _ p , 1 − 1 + 3 p ) + ∥ N 0 − P 0 ∥ B _ q , 1 − 2 + 3 q ) exp { C 0 ( ∥ N 0 + P 0 ∥ B _ r , 1 − 2 + 3 r + 1 ) exp { C 0 ∥ u 0 ∥ B _ p , 1 − 1 + 3 p } } ≤ c 0 , then the system admits a unique global solution, where u L = e t ∆ u 0 , and e t ∆ denotes the heat flow. In addition, by using the weighted Chemin-Lerner type norm and interpolation techniques, we establish the optimal temporal decay rates of these global large solutions.

In this paper, we propose a coupled system that describes the interaction between the Relativistic-Cucker-Smale model and the incompressible Navier-Stokes equations by means of a drag force and establish a theory of global existence, as well as the time-asymptotic behavior of the proposed model in T 3 . It is shown that the coupled system exhibits an exponential alignment under some specific assumptions, and that weak solutions exist globally for general initial data.

It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain Ω ⊂ R 2 : { u t = ∆ u − χ ∇ · ( u v k ∇ v ) + ru − µ u 2 , in Ω × ( 0 , T max ) , v t = ∆ v − αv + βu , in Ω × ( 0 , T max ) , where k∈(0 ,1), and χ,r,µ,α,β are positive parameters. Known results have already established the same conclusion for the parabolic-elliptic case. Here, we complement these findings by extending the result to the fully parabolic case.

This paper examines the critical exponents for the existence of global solutions to the equation u t − ∆ H u = ∫ 0 t ( t − s ) − γ | u ( s ) | p − 1 u ( s ) ds , 0 ≤ γ < 1 , η ∈ H n , t > 0 , on the Heisenberg groups H n . There exists a critical exponent p c = max { 1 γ , n + 2 n + γ } ∈ ( 0 , + ∞ ] , such that for all 1 < p ≤ p c , no global solution exists regardless of the non-negative initial data, while for p > p c , a global positive solution exists if the initial data is sufficiently small. The results obtained are a natural extension of the results of Cazenave et al. [Nonlinear Analysis 68 (2008), 862-874], where similar studies were carried out in R n . Furthermore, several theorems are presented that provide lifespan estimates for local solutions under various initial data conditions. The proofs of the main results are based on test function methods and Banach fixed point principle.

A spectral parameter power series (SPPS) representation for the regular solution of the radial Dirac system with complex coefficients is obtained, as well as a SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. Based on the SPPS representation, a numerical method for solving spectral problems is developed. It is shown that the method is also applicable to solving spectral problems for perturbed Bessel equations. We exhibit that the proposed numerical method delivers excellent results. Additionally, an application of the method to find the energy values of an electron orbiting a hydrogen-like atom with a finite radius is presented.