Homogeneous Neumann Boundary Conditions Research Articles

Global Existence and Fixed-Time Synchronization of a Hyperchaotic Financial System Governed by Semi-Linear Parabolic Partial Differential Equations Equipped with the Homogeneous Neumann Boundary Condition

Chaotic nonlinear dynamical systems, in which the generated time series exhibit high entropy values, have been extensively used and play essential roles in tracking accurately the complex fluctuations of the real-world financial markets. We are concerned with a system of semi-linear parabolic partial differential equations supplemented by the homogeneous Neumann boundary condition, which governs a financial system comprising the labor force, the stock, the money, and the production sub-blocks distributed in a certain line segment or planar region. The system derived by removing the terms involved with partial derivatives with respect to space variables from our concerned system was demonstrated to be hyperchaotic. We firstly prove, via Galerkin's method and establishing a priori inequalities, that the initial-boundary value problem for the concerned partial differential equations is globally well posed in Hadamard's sense. Secondly, we design controls for the response system to our concerned financial system, prove under some additional conditions that our concerned system and its controlled response system achieve drive-response fixed-time synchronization, and provide an estimate on the settling time. Several modified energy functionals (i.e., Lyapunov functionals) are constructed to demonstrate the global well-posedness and the fixed-time synchronizability. Finally, we perform several numerical simulations to validate our synchronization theoretical results.

Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result

We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse interface has developed and intersects the boundary ∂ Ω. The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [ 0 , T ] for some time T > 0. Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).

On the existence of solutions for a parabolic‐elliptic chemotaxis model with flux limitation and logistic source

In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ ” and a chemical stimulus “ ” in a bounded and regular domain of . The equation for is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as for . The chemical substance distribution satisfies the elliptic equation The evolution of is also determined by a logistic type growth term . The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for and any .

On reaction-diffusion models with memory and mean curvature-type diffusion

We propose and study a reaction-diffusion model with memory, where the linear diffusion operator is replaced by the mean curvature operator both in Euclidean and Lorentz–Minkowski spaces and the memory kernel is assumed to be of Jeffreys type. Regarding the reaction term, we consider a balanced bistable function f=−F′, with F a generic double well potential with wells of equal depth. In particular, we assume that the potential F has two global minima at ±1 and that F(u)∼|1±u|2+θ, for some θ>−1, when u≈±1, and we consider the corresponding equation in a bounded interval with homogeneous Neumann boundary conditions. We prove that if θ∈(−1,0), then there exist special steady states, named compactons, with a transition layer structure. In contrast, if θ≥0 the interface layers are not stationary and two different phenomena emerge: for θ=0 solutions exhibit a metastable behavior and maintain an unstable structure for an exponentially long time, while if θ>0 the exponentially slow motion is replaced by an algebraic one.

On the global existence of solutions to chemotaxis system for two populations in dimension two

We consider the global existence for the following fully parabolic chemotaxis system with two populations \[\left\{ \begin{array}{@{}ll} \partial_tu_i=\kappa_i\Delta u_i-\chi_i\nabla\cdot(u_i\nabla v),\quad i\in\{1,2\}, & x\in\Omega,\ t>0, \\ v_t=\Delta v-v+u_1+u_2, & x\in\Omega,\ t>0,\\ u_i(x,t=0)=u_{i0}(x),\quad v(x,t=0)=v_0(x), & x\in\Omega, \end{array} \right. \]where $\Omega =\mathbb {R}^2$ or $\Omega =B_R(0)\subset \mathbb {R}^2$ supplemented with homogeneous Neumann boundary conditions, $\kappa _i,\chi _i>0,$$i=1,2$. The global existence remains open for the fully parabolic case as far as the author knows, while the existence of global solution was known for the parabolic-elliptic reduction with the second equation replaced by $0=\Delta v-v+u_1+u_2$ or $0=\Delta v+u_1+u_2$. In this paper, we prove that there exists a global solution if the initial masses satisfy the certain sub-criticality condition. The proof is based on a version of the Moser–Trudinger type inequality for system in two dimensions.

Complex dynamics induced by harvesting rate and delay in a diffusive Leslie-Gower predator-prey model

<abstract><p>In this paper, under homogeneous Neumann boundary conditions, the complex dynamical behaviors of a diffusive Leslie-Gower predator-prey model with a ratio-dependent Holling type III functional response and nonlinear prey harvesting is carefully studied. By scrupulously analyzing and comprehending the distribution of the eigenvalues, the existence and stability (balance) of the extinction and coexistence equilibrium states are determined, and the bifurcations exhibited by the system are investigated by a mathematical analysis. Additionally, based on the theoretical analysis and numerical simulation, (Harvesting rate-induced, Delay-induced), Turing-Hopf bifurcations points are derived. Our results show that delay and nonlinear prey harvesting rates can create spatially inhomogeneous periodic solutions.</p></abstract>

Boundedness and asymptotic behavior in a quasilinear two-species chemotaxis system with loop

Chemotaxis is the directed movement of cells or organisms in response to the gradients of concentration of the chemical stimuli, plays essential roles in various biological process. In this paper, we consider a quasilinear two-species chemotaxis-competition system with loop in higher dimensions defined on a smooth bounded domain with homogeneous Neumann boundary conditions. With the help of suitable energy functional and some estimates, we obtain the global existence and boundedness of the classical solution of the model, and also show the large time behavior and convergence rate of the solution. Our results show that the nonlinear diffusion mechanism has an inhibitory effect on the blow-up of solution and partially generalized some results in the literature.

Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth

<p style='text-indent:20px;'>We provide explicit examples of finite time <inline-formula><tex-math id="M1">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-blow up for the solutions of <inline-formula><tex-math id="M2">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> reaction-diffusion systems for which three main properties hold: positivity is preserved for all time, the total mass is uniformly controlled and the growth of the nonlinear reaction terms is superquadratic. They are obtained by choosing the space dimension large enough. This is to be compared with recent global existence results of uniformly bounded solutions for the same kind of systems with quadratic or even slightly superquadratic growth depending on the dimension. Such blow up may occur even with homogeneous Neumann boundary conditions. All these <inline-formula><tex-math id="M3">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>-blowing up solutions may be extended as weak global solutions. Blow up examples are also provided in space dimensions one, two and three with various growths.</p>

Qualitative analysis on the diffusive Holling–Tanner predator–prey model

We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steady-state. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel'kov model to obtain the non-existence of positive non-constant steady-states if 0 < p ≤ 1 .

Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

<abstract><p>In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.</p></abstract>

Global dynamics for a chemotaxis consumption system with signal-dependent motility and logistic source

This paper deals with the chemotaxis consumption model{ut=Δ(uϕ(v))+au−buγ,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1). Here the parameters a>0, b>0 and γ>1, and the motility function ϕ satisfies ϕ∈C3([0,∞)) with ϕ(s)>0 for all s≥0. For all suitably regular initial data, if one of the following cases holds: (i) n≤2, γ>1; (ii) n≥3, γ>2; (iii) n≥3, γ=2 and b is large, then this system possesses global bounded classical solutions, while if n≥3 and γ∈(1,2], then this system admits at least one global weak solution, which becomes smooth after some waiting time and which will converge to the constant equilibrium ((ab)1γ−1,0).

Global boundedness and asymptotics of a class of prey‐taxis models with singular response

This paper is concerned with a class of singular prey‐taxis models in a smooth bounded domain under homogeneous Neumann boundary conditions. The main challenge of analysis is the possible singularity as the prey density vanishes. Employing the technique of a priori assumption, the comparison principle of differential equations and semigroup estimates, we show that the singularity can be precluded if the intrinsic growth rate of prey is suitably large and hence obtain the existence of global classical bounded solutions. Moreover, the global stability of co‐existence and prey‐only steady states with convergence rates is established by the method of Lyapunov functionals.

Null controllability for a structurally damped stochastic plate equation

In this paper, we obtain a null controllability result for a structurally damped refined stochastic plate equation with two kinds of boundary conditions. The key point is to establish an observability inequality for the backward structurally damped stochastic plate equation. For this purpose, we derive a new global Carleman estimate for the damped stochastic plate operator with homogeneous Dirichlet boundary conditions on the solution and the laplacian of the solution and with homogeneous Neumann boundary conditions of first and third order, respectively. Such an estimate is obtained by using the Carleman estimate for stochastic parabolic operators with complex coefficients in an iteration manner.

The existence and stability of spikes in the one-dimensional Keller-Segel model with logistic growth.

It is well known that Keller-Segel models serve as a paradigm to describe the self aggregation phenomenon, which exists in a variety of biological processes such as wound healing, tumor growth, etc. In this paper, we study the existence of monotone decreasing spiky steady state and its linear stability property in the Keller-Segel model with logistic growth over one-dimensional bounded domain subject to homogeneous Neumann boundary conditions. Under the assumption that chemo-attractive coefficient is asymptotically large, we construct the single boundary spike and next show this non-constant steady state is locally linear stable via Lyapunov-Schmidt reduction method. As a consequence, the multi-symmetric spikes are obtained by reflection and periodic extension. In particular, we present the formal analysis to illustrate our rigorous theoretical results.

Global Stability for a Diffusive Infection Model with Nonlinear Incidence

The first purpose of this article was to establish and analyze system 4 with an abstract function incidence rate under homogeneous Neumann boundary conditions. The system models the dynamics of interactions between pathogens and the host immune system, which has important applications in HIV-1, HCV, flu etc. By the Lyapunov–LaSalle method, we obtained the globally asymptotic stability of the equilibria. Specifically speaking, by introducing the reproductive numbers R0 and R1, we showed that if R0≤1, then the infection-free equilibrium E0 is globally asymptotically stable, i.e., the virus is unable to sustain the infection and will become extinct; if R1≤1<R0, then the CTL-inactivated infection equilibrium E1 is globally asymptotically stable, i.e., the infection becomes chronic but without persistent CTL response; if R1>1, the CTL-activated equilibrium E2 is globally asymptotically stable, and the infection is chronic with persistent CTL response. Additionally, we also investigate the discretization of the model by using a non-standard finite difference scheme, and our results confirm that the global stability of the equilibria of the continuous model and the discrete model is consistent. Finally, numerical simulations are performed to illustrate the theoretical results. Our model and results are to a certain extent generalizations of and improvements upon the previous results given by Zhu, Wang.

Level-set forced mean curvature flow with the Neumann boundary condition

Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

Численное моделирование взаимодействия газовзвеси с ударной волной континуальными математическими моделями с идеальной и диссипативными несущими средами

This paper compares computer implementations of numerical algorithms for solving the equations of mathematical models of the dynamics of gas suspensions with viscous heat-conducting, inviscid heatconducting and ideal carrier media. Mathematical models are developed within the framework of the continuum technique for modeling the dynamics of multiphase media. In the study, the process of interaction of a shock wave moving from a homogeneous gas into a gas suspension, which is often encountered in the mining industry, was modeled. The relevance of the study of this flow of inhomogeneous media is associated with the shielding of industrial explosions by aerosol curtains. When modeling for a viscous medium, homogeneous Dirichlet boundary conditions were set, for an inviscid medium, homogeneous Neumann boundary conditions. The equations of the mathematical model were integrated by the McCormack finite difference method. To overcome numerical oscillations, a nonlinear scheme for correcting grid functions was used. The program that implements the continuum method for the dynamics of multiphase media consisted of a block for specifying boundary conditions, a block that implements a numerical solution, and a block for accounting for interfacial interaction. As a result of comparing numerical calculations of mathematical models of the dynamics of a gas suspension with an ideal, inviscid heat-conducting and viscous heat-conducting carrier media, it was found that during the movement of a gas suspension, the viscosity of the carrier medium of the gas suspension has the greatest influence on the intensity of interfacial momentum exchange.

Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System

In this paper, we consider the following indirect signal generation and logarithmic sensitivity ε n t = Δ n − χ ∇ · n ∇ ln c x ∈ Ω , t > 0 c t = Δ c − c + w x ∈ Ω , t > 0 w t = Δ w − w + n x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a ball domain Ω ⊂ R N N ≥ 4 with smooth boundary ∂ Ω . This paper considers in the singular limit ε ⟶ 0 ; the result comes from the finite time blow-up of arbitrary large values of n in the corresponding nonlocal scalar parabolic equation case when N ≥ 4 and χ > 2 N / N − 2 .

A REACTION-ADVECTION-DIFFUSION MODEL IN SPATIAL ECOLOGY: THEORETICAL AND COMPUTATIONAL ANALYSIS

In a confined heterogeneous habitat with two species interacting for common resources, the research analyzes a reaction-advection-diffusion type dispersal model with homogeneous Neumann boundary conditions for generalized growth functions. Both species follow the same symmetric growths law, but their dispersal strategies and advection rates are different. The following pattern is used to consider the competition strategy: in a bounded heterogeneous habitat, the first population disperses according to its resource functions, whereas the second population disperses according to its carrying capacity. We investigate the model in two scenarios: when carrying capacity and resource functions are non-proportional, competitive exclusion occurs, and one species drives theother to extinction in the long run for various similar and unequal carrying capacities of competing species. However, coexistence is achievable for different resource distribution consumption if the resource distribution and the carrying capacity phase of the second species are non-constant and similar. A series of numerical computations are used to demonstrate the model’s efficacy in oneand two-dimensional instances, which is particularly significant for environmental consideration.

A Boundary Shape Function Method for Computing Eigenvalues and Eigenfunctions of Sturm–Liouville Problems

In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these two canonical forms. Owing to the property of the boundary shape function, we could transform the SLPs into an initial value problem for the new variable with initial values that were given definitely. Meanwhile, the terminal value at the right boundary could be entirely determined by using a given normalization condition for the uniqueness of the eigenfunction. In such a manner, we could directly determine the eigenvalues as the intersection points of an eigenvalue curve to the zero line, which was a horizontal line in the plane consisting of the zero values of the target function with respect to the eigen-parameter. We employed a more delicate tuning technique or the fictitious time integration method to solve an implicit algebraic equation for the eigenvalue curve. We could integrate the Sturm–Liouville equation using the given initial values to obtain the associated eigenfunction when the eigenvalue was obtained. Eight numerical examples revealed a great advantage of the BSFM, which easily obtained eigenvalues and eigenfunctions with the desired accuracy.