Homogeneous Neumann Boundary Conditions Research Articles

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Asymptotic Behavior of Global Solution of an Anomalous Gray Scott Model

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. However, the asymptotic behavior is investiged by technique semi group estimates and the Sobolev embedding theorem      

Solutions to a coupled Cahn–Hilliard-phase-field-crystal model for grain boundary motion

In this paper, we consider an initial–boundary value problem (IBVP) of a coupled Cahn–Hilliard-phase-field-crystal (CH-PFC) system subject to homogeneous Neumann boundary conditions in one spatial dimension. This CH-PFC model couples the composition field of a diffusing species with the crystallographic and can be used to model the diffusion-induced grain boundary migration in crystalline materials. Under suitable assumptions on the coefficients and initial data, we prove that the IBVP possesses a global weak solution. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension.

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

Steady States of a Diffusive Population-Toxicant Model with Negative Toxicant-Taxis

This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4).

Stability of Solutions to a Caginalp Phase-Field Type Equations

This paper is concerned with the study of the asymptotic behavior of a generalization of the Caginalp phase-field model subject to homogeneous Neumann boundary conditions and regular potentials involving two temperatures. This work follows on from a paper in which the well-posedness of the problem, the dissipativity of the system, and the existence of global and exponential attractors were demonstrated. In addition, a study on the semi-infinite cylinder was also carried out. Indeed, if it is true that the existence of a global attractor makes it possible to predict the asymptotic behavior of solutions on a bounded domain, it does not say that these solutions converge. After having shown the existence of the global attractor, it is therefore important to look at the convergence of the solutions over time. There are several methods for determining the asymptotic behavior of the solutions of a differential system. We can mention the one that consists of transforming the given differential equations into integral equations and then applying the classical Picard successive approximation procedure to them. This work is devoted to the study of the convergence of solutions to steady states, adapting a well-known result concerning Lojasiewicz-Simon’s inequality.

Global boundedness and asymptotic stabilization in a chemotaxis system with density-suppressed motility and nonlinear signal production

In this paper, we study the following chemotaxis model with density-suppressed motility and nonlinear production{ut=Δ(φ(v)u)+ru−μuα,x∈Ω,t>0,vt=Δv−v+wβ,x∈Ω,t>0,wt=Δw−w+uγ,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn(n≥2) with smooth boundary, where r∈R,μ,β,γ>0 and α>1. The positive motility function satisfies φ(s)∈C3([0,∞)) and φ′(s)≤0 for all s≥0. It is showed that the system admits a globally bounded and classical solution under some conditions on α,β and γ. Then, under stricter constraints on φ, we obtained that the parameter β has a wider range than before, which is enough to ensure the global boundedness of the solution. Furthermore, if μ is sufficiently large, we proved that the solution converges to ((r+μ)1α−1,(r+μ)βγα−1,(r+μ)γα−1) in L∞(Ω) as t→∞ for all r∈R,μ,β,γ>0 and α>1, where r+:=max⁡{r,0}. Finally, we showed that the convergence is exponential in cases r>0 and r<0.

Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

Turing–Turing bifurcation in an activator–inhibitor system with gene expression time delay

In this paper, the codimension-two Turing–Turing bifurcation of an activator–inhibitor system with gene expression time delay is investigated under the homogeneous Neumann boundary condition. The interaction between two different Turing modes gives rise to the Turing–Turing bifurcation, leading to the emergence of multi-stable and superposed spatial patterns. Rigorous theoretical analysis is first given to study the Turing, Hopf, and Turing–Turing bifurcations of this system. Subsequently, in order to describe the spatiotemporal dynamics resulting from the Turing–Turing bifurcation in more detail, the algorithm for calculating the third-order truncated normal form of the Turing–Turing bifurcation is derived using the center manifold theorem and normal form theory. This algorithm can be applied to analyze the Turing–Turing bifurcation in other diffusive systems with gene expression time delay. The derived normal form enables the theoretical prediction of the spatiotemporal dynamics of this system near the Turing–Turing bifurcation point. Numerical simulations are conducted to support the theoretical analysis, revealing the presence of superposition patterns and quad-stable patterns in particular.

On small-data solution of the chemotaxis–SIS epidemic system with bilinear incidence rate

This paper deals with the initial–boundary value problem for chemotaxis susceptible-infected-susceptible epidemic system with bilinear incidence rate ut=d1Δu+χ∇⋅(u∇v)−β(x)uv+γ(x)v,x∈Ω,t>0,vt=d2Δv+β(x)uv−γ(x)v,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0, χ∈R, 0<β∈C1(Ω¯) and 0<γ∈C1(Ω¯). It is proved that if (u(x,0),v(x,0)) in L∞(Ω)×L1(Ω) is suitably small and ‖v(x,0)‖L∞(Ω)+‖∇v(x,0)‖Lq(Ω)≤K for each K>0 and q>n, then the problem possesses a unique global classical solution which is bounded in Ω×(0,∞). Moreover, we prove that the solution exponentially stabilizes to the disease-free equilibrium N|Ω|,0 with N=∫Ω(u(x,0)+v(x,0)) in L∞(Ω) as t→∞.

Bifurcation and Spatiotemporal Patterns of SI Epidemic Model with Diffusion

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations.

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.&lt;/p&gt;&lt;/abstract&gt;

Dynamic behavior analysis to a generalized chemotaxis-consumption system

In this paper, the following chemotaxis system with nonlinear consumption mechanisms is considered ut = Δu − χ∇ · (u∇v) + ξ∇ ·  (u∇w) + au − bum, vt = Δv − uαv, wt = Δw − uβw under homogeneous Neumann boundary conditions, where Ω⊂Rn(n≥2) is a smoothly bounded domain and parameters χ, ξ, a, b, α, β &amp;gt; 0 and m &amp;gt; 1. If m and l = max{α, β} satisfy m&amp;gt;maxl(n+2)2,1, then the system possesses a global classical solution, which is bounded in Ω × (0, ∞). Furthermore, it has been shown that such solution exponentially converges to equilibrium ((ab)1m−1,0,0) as t → ∞, where convergence rate can be formally characterized by the parameters of the system.

Dynamics of a diffusive two predators- one prey system

This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.

Complexity for a class of elliptic ordinary integro-differential equations

Consider the variational form of the ordinary integro-differential equation (OIDE)−u″+u+∫01q(⋅,y)u(y)dy=f on the unit interval I, subject to homogeneous Neumann boundary conditions. Here, f and q respectively belong to the unit ball of Hr(I) and the ball of radius M1 of Hs(I2), where M1∈[0,1). For ε>0, we want to compute ε-approximations for this problem, measuring error in the H1(I) sense in the worst case setting. Assuming that standard information is admissible, we find that the nth minimal error is Θ(n−min⁡{r,s/2}), so that the information ε-complexity is Θ(ε−1/min⁡{r,s/2}); moreover, finite element methods of degree max⁡{r,s} are minimal-error algorithms. We use a Picard method to approximate the solution of the resulting linear systems, since Gaussian elimination will be too expensive. We find that the total ε-complexity of the problem is at least Ω(ε−1/min⁡{r,s/2}) and at most O(ε−1/min⁡{r,s/2}ln⁡ε−1), the upper bound being attained by using O(ln⁡ε−1) Picard iterations.

Blowup in L 1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

Abstract This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity u p {u}^{p} in a bounded domain Ω \Omega with the homogeneous Neumann boundary condition and positive initial values. In the case of p &gt; 1 p\gt 1 , we prove the blowup of solutions u ( x , t ) u\left(x,t) in the sense that ‖ u ( ⋅ , t ) ‖ L 1 ( Ω ) \Vert u\left(\hspace{0.33em}\cdot \hspace{0.33em},t){\Vert }_{{L}^{1}\left(\Omega )} tends to ∞ \infty as t t approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of 0 &lt; p &lt; 1 0\lt p\lt 1 , we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

Global classical solvability and asymptotic behaviors of a parabolic-elliptic Chemotaxis-type system modeling crime activities

This paper is devoted to a two-dimensional parabolic-elliptic system in crime activities. The main difference between this system and the chemotaxis system with logarithmic singular sensitivity and quadratic logistic source is that the source term of this system is a mixtype quadratic term. The local existence of solutions to its intial boundary valve problem associated with homogeneous Neumann boundary conditions was considered by Rodríguez. The present study indicates that for any properly regular initial data there admits global classical solutions. Moreover, the qualitative behaviors of solutions in large time scales are considered as well.

Quality of approximating a mass-emitting object by a point source in a diffusion model

For the sake of computational efficiency and for theoretical purposes, in mathematical modelling, the Dirac Delta distributions are often utilized as a replacement for cells or vesicles, since the size of cells or vesicles is much smaller than the size of the surrounding tissues. Here, we consider the scenario that the cell or the vesicle releases the diffusive compounds to the immediate environment, which is modelled by the diffusion equation. Typically, one separates the intracellular and extracellular environment and uses homogeneous Neumann boundary condition for the cell boundary (so-called spatial exclusion model), while the point source model neglects the intracellular environment. We show that extra conditions are needed such that the solutions to the two models are consistent. We prove a necessary and sufficient condition for the consistency. Suggested by the numerical results, we conclude that an initial condition in the form of Gaussian kernel in the point source model compensates for a time-delay discrepancy between the solutions to the two models in the numerical solutions. Various approaches determining optimal amplitude and variance of the Gaussian kernel have been discussed.