Homogeneous Boundary Conditions Research Articles

Coexistence states for a prey-predator model with self-diffusion and predator-taxis

This paper is concerned with the stationary problem of a prey-predator model with self-diffusion and predator-taxis. Under homogeneous Dirichlet boundary conditions, the bifurcation approach is used to derive the sufficient conditions for the existence of coexistence states. Our analysis shows that the coupling of self-diffusion and predator-taxis cause some difficulties in establishing a priori estimates for the solutions.

Stabilization for degenerate equations with drift and small singular term

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Numerical Study of the Influence of Boundary Conditions on Calculations of the Dynamics of Polydisperse Gas Suspension

The work numerically simulates the flow of a polydisperse gas suspension in a channel. The carrier medium was described as a viscous, compressible, heat-conducting gas. The mathematical model implemented a continuum technique for the dynamics of multiphase media, taking into account the interaction of the carrier medium and the dispersed phase. For each component of the mixture, a complete hydrodynamic system of equations of motion for the carrier phase and dispersed phase fractions was solved. The dispersed phase consisted of particles with different sizes of dispersed inclusions. For the carrier medium, homogeneous Dirichlet boundary conditions were specified on the side surfaces of the channel. For fractions of the dispersed phase, boundary conditions for slippage. The influence of the boundary conditions of the flow of the carrier medium on the dynamics of gas suspension fractions has been revealed.

Asymptotic stability in a predator-prey system with density-dependent diffusion and indirect pursuit-evasion interaction in 2D

This paper deals with a predator-prey system with density-dependent diffusion and indirect pursuit-evasion interaction under homogeneous Neumann boundary conditions. By providing appropriate conditions, the asymptotic stability of solution is investigated.

Absence of dead-core formations in chemotaxis systems with degenerate diffusion

In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form ut=∇⋅(D(u)∇u−uS(u)∇v)+f(u,v),vt=Δv−uv,in a bounded domain Ω⊂RN with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient D∈C0([0,∞))∩C2((0,∞)) is assumed to satisfy D(0)=0, D(s)>0 on (0,∞), D′(s)≥0 on (0,∞) and that there are s0>0, p>1 and CD>0 such that sD′(s)≤CDD(s)andCDsp−1≤D(s)fors∈[0,s0].The sensitivity function S∈C2([0,∞)) and the source term f∈C1([0,∞)×[0,∞)) are supposed to be nonnegative.We show that for all suitably regular initial data (u0,v0) satisfying u0≥δ0>0 and v0⁄≡0 there is a time-local classical solution and – despite the degeneracy at 0 – the solution satisfies an extensibility criterion of the form eitherTmax=∞,orlim supt↗Tmax‖u(⋅,t)‖L∞(Ω)=∞.Moreover, as a by-product of our analysis, we prove that a classical solution on Ω×(0,T) obeying ‖u(⋅,t)‖L∞(Ω)≤Mu for all t∈(0,T) and emanating from initial data (u0,v0) as specified above remains strictly positive throughout Ω×(0,T), i.e. one can find δu=δu(T,δ0,Mu,‖v0‖W1,∞(Ω))>0 such that u(x,t)≥δufor all(x,t)∈Ω×(0,T).Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time.

A study of sign-changing Poisson-type equations in two configurations

Abstract. In this paper, We are interested in specific non-coercive issues concerning electromagnetic wave propagation in the presence of metals or particular metamaterials. We focus on some non-coercive problems which cannot be studied by a classical Lax-Milgram approach. We consider the sign-changing Poisson-type equations with the homogeneous Dirichlet boundary conditions in the circular configuration and in the three square configuration. We utilize a method called T-isomorphism, which allows the conversion of non-coercive problems into coercive ones, to examine specific Poisson-type equations with sign-changing parameters. We first use the classical method to transform the equations into the corresponding variational formulations, and then apply the T-isomorphism method to show the well-posedness of these variational formulations with some parameters. By applying appropriate isomorphisms, we can conclude the well-posedness of the related variational formulations with certain parameters, which constitute the main findings of this paper. After giving appropriate isomorphisms, we can deduce the well-posedness of the corresponding variational formulations with some parameters, which are our main results in this paper.

Free vibration analysis of rectangular plates resting on an elastic foundation

Abstract Rectangular plate on elastic foundations is a common structural form in aerospace fields, while there still lacks a general analytical method for free vibration analysis of this kind of structure. Thus, the extended separation-of-variable method is used to obtain the analytical free vibration solutions for plates with all kinds of homogeneous boundary conditions. The mode functions are assumed to be in separation-of-variable form, and the natural frequencies of two-direction mode functions are mathematically independent. Then, the free vibration solutions are obtained according to the two-direction boundary conditions. The correctness of the present results is verified via numerical comparisons. Parameter study shows that the elastic foundation significantly affects the flexural vibration behaviors of rectangular plates, and the influence is dependent on boundary conditions and mode orders of rectangular plates.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

Global existence and boundedness in a two‐dimensional parabolic chemotaxis system with competing attraction and repulsion effects

In this paper, we investigate a chemotaxis system under homogeneous Neumann boundary conditions within a bounded domain with a smooth boundary. The system describes the movement of cells in response to two chemical signal substances: one acts as a chemoattractant, while the other serves as a chemorepellent, both produced by the cells. The system takes into account chemotactic sensitivity in the reaction movement when detecting these chemicals. Under certain assumptions, we demonstrate the existence of a unique global bounded classical solution for the proposed problem. To further understand the time evolution of the system's solutions, we conduct numerical experiments and analyze the dynamic properties of the norm of the solutions with respect to variations in chemical production rates.

Boundary vorticity of incompressible 2D flows

For a homogeneous incompressible 2D fluid confined within a bounded Lipschitz simply connected domain, homogeneous Neumann pressure boundary conditions are equivalent to a constant boundary vorticity. We investigate the rigidity of such conditions.

Dynamic behavior in a pursuit-evasion system with signaling mechanism

We mainly focus on the predator-prey problem associated with indirect predator taxis{ut=Δu+χ1∇⋅(u∇w)+u(1−μ1u−αv),vt=Δv−χ2∇⋅(ψ(v)∇u)+v(−1−μ2v+βu),0=Δw+v−w in a smooth bounded domain Ω⊂Rn(n≤3) with homogeneous Neumann boundary conditions, where the parameters χ1, χ2, μ1, μ2, α and β are positive. The chemotaxis sensitivity function with/without “volume-filling” effect ψ(v)=v1+σv, where σ≥0 is a constant. Compared with the well-known predator-prey system, the discussion of signaling mechanism w forms a novel feature of our system. For the case σ=0, we identify a positive constant χ0 such that the above system possesses global classical solutions with relative bound if χ2<χ0; For the case σ>0, we also show that the classical solutions of the system are globally bounded without any restrictions on system parameters. Furthermore, utilizing the Lyapunov functionals, we establish stability for the prey-only situation, the steady state of coexistence is also investigated.

Effective models of metasurface/metainterface made of three-dimensional Helmholtz resonators

We study a reflective metasurface made of 3D subwavelength Helmholtz resonators. We provide an effective model in the time domain obtained by using a three-scale asymptotic approach. The metasurface is reduced to an homogenized boundary condition involving a resonant, frequency dependant impedance. Our effective model takes into account boundary layer effects near the neck opening, and we provide a precise method to calculate the " added length ". The study is extended to meta-interface in transmission, where the set of effective boundary conditions are replaced by effective jump conditions, and the results are discussed in frequency domain with comparison with direct numerical simulations.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity

This article examines the weak solution of a fully parabolic chemotaxis-competition system with loop and signal-dependent sensitivity. The system is subject to homogeneous Neumann boundary conditions within an open, bounded domain \(\Omega\subset\mathbb{R}^n\), where \(n\geq 1\) and \(\partial\Omega\) is smooth. We assume that the parameters in the system are positive constants. Additionally, the initial data \((u_{10}, u_{20}, v_{10}, v_{20})\in L^2(\Omega)\times L^2(\Omega) \times W^{1,2}(\Omega)\times W^{1,2}(\Omega)\) are non-negative. The existence of a weak solution to the problem is established using energy inequality method. For more information see https://ejde.math.txstate.edu/Volumes/2024/56/abstr.html

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

Nonexistence for a system of time‐fractional diffusion equations in an exterior domain

A system of time‐fractional diffusion equations posed in an exterior domain of ( ) under homogeneous Dirichlet boundary conditions is investigated in this paper. The time‐fractional derivatives are considered in the Caputo sense. Using nonlinear capacity estimates specifically adapted to the nonlocal properties of the Caputo fractional derivative, the geometry of the domain, and the boundary conditions, we obtain sufficient conditions for the nonexistence of a weak solution to the considered system.

Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting

This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions.

Existence of global weak solutions and simulations to a Dirichlet problem for a generalized Swift–Hohenberg equation

In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form ψ2Δ2ψ2 in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter β<0, where we employ the energy method and make use of various techniques to derive delicate a priori estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.

Dynamics of a nonlinear infection viral propagation model with one fixed boundary and one free boundary

In this paper we study a nonlinear infection viral propagation model with diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet boundary conditions, while the right boundary is free. We find that the habitat always expands to the half line [0,∞), and that the virus and infected cells always die out when the Basic Reproduction NumberR0≤1, while the virus and infected cells have persistence properties when R0>1. To obtain the persistence properties of virus and infected cells when R0>1, the most work of this paper focuses on the existence and uniqueness of bounded positive equilibrium solutions for subsystems and the existence of positive equilibrium solutions for the entire system.