Homogeneous Boundary Research Articles

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO 2D G-BÉNARD PROBLEM IN UNBOUNDED DOMAINS

We consider the 2D g-Bénard problem in domains satisfying the Poincaré inequality with homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of global weak solutions. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Bénard problem.

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

AbstractLet $\Omega \subset \mathbb {R}^N$ , $N\geq 2$ , be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta )^s u =\lambda \rho u$ in $\Omega $ with homogeneous Dirichlet boundary condition, where $(-\Delta )^s$ , $s\in (0,1)$ , is the fractional Laplacian operator, $\lambda \in \mathbb {R}$ and $ \rho \in L^\infty (\Omega )$ .We study weak* continuity, convexity and Gâteaux differentiability of the map $\rho \mapsto 1/\lambda _1(\rho )$ , where $\lambda _1(\rho )$ is the first positive eigenvalue. Moreover, denoting by $\mathcal {G}(\rho _0)$ the class of rearrangements of $\rho _0$ , we prove the existence of a minimizer of $\lambda _1(\rho )$ when $\rho $ varies on $\mathcal {G}(\rho _0)$ . Finally, we show that, if $\Omega $ is Steiner symmetric, then every minimizer shares the same symmetry.

The spectrum of the mean curvature operator

AbstractWe show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

Equilibrium stability of nonlinear elastic sphere with distributed dislocations

An exact formulation and a solution of the stability problem for a three-dimensional elastic body containing distributed dislocations are given. The buckling phenomenon for a hollow nonlinear elastic sphere of the semi-linear (harmonic) material with edge dislocations is studied. The study is carried out within the framework of the continuum theory of continuously distributed dislocations using the bifurcation method of buckling analysis. The bifurcation method is to find the equilibrium positions of an elastic body, which differ little from the subcritical (unperturbed) state. The perturbed equilibrium state is described by a linearized boundary value problem. By solving a homogeneous linear boundary value problem, the minimum critical value of the external pressure at which the sphere loses stability is found. The influence of dislocations on the buckling of both thin and thick spherical shells is analyzed.

Extinction of solutions in parabolic equations with different diffusion operators

ABSTRACT In this paper, we study the evolution p, q-Laplacian equations and with 1<p, q<2, subject to homogeneous Dirichlet boundary conditions. If , there exist suitable initial data such that vanishing solutions exist. If , we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case , the solutions vanish in finite time with small initial data.

Global dynamics of a diffusive viral infection model with general incidence function and distributed delays

The distributed delay was firstly proposed by Volterra in the 1930s since it is more realistic than discrete delay and has been introduced in many dynamical systems. In this paper, we establish a diffusive viral infection model with general incidence function and distributed delays subject to the homogeneous Neumann boundary conditions. Firstly, we prove the existence, uniqueness, positivity and boundedness of solutions of the model. Then, by using the linearization method and constructing appropriate Lyapunov functionals, we show that the global dynamics of the model is determined by the reproductive numbers for viral infection $$\mathcal {R}_{0}$$ , which implies that the global stability of the model precludes the existence of complex dynamical behaviors such as Hopf bifurcation and patter formation. Furthermore, an example is presented and numerical simulations are also carried out to illustrate the main results.

Boundedness in a two-species chemotaxis-consumption system with nonlinear diffusion and sensitivity

ABSTRACT This paper deals with the following two-species chemotaxis system with homogeneous Neumann boundary conditions and suitable initial conditions, where () is a bounded and smooth domain. Moreover, the parameters , and . For i = 1, 2, we assume that where , , and . Then if one of the following cases holds: ; and is sufficiently large, it is proved that the system possesses a unique global bounded classical solution.

The Stochastic Viscous Cahn\u2013Hilliard Equation: Well-Posedness, Regularity and Vanishing Viscosity Limit

Well-posedness is proved for the stochastic viscous Cahn–Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular, also super-polynomial) provided that it is everywhere defined on the real line. A vanishing viscosity argument is carried out and the convergence of the solutions to the ones of the pure Cahn–Hilliard equation is shown. Some refined regularity results are also deduced for both the viscous and the non-viscous case.

Dynamics of a diffusive predator–prey model with herd behavior

This paper is devoted to considering a diffusive predator–prey model with Leslie–Gower term and herd behavior subject to the homogeneous Neumann boundary conditions. Concretely, by choosing the proper bifurcation parameter, the local stability of constant equilibria of this model without diffusion and the existence of Hopf bifurcation are investigated by analyzing the distribution of the eigenvalues. Furthermore, the explicit formula for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are also derived by applying the normal form theory. Next, we show the stability of positive constant equilibrium, the existence and stability of periodic solutions near positive constant equilibrium for the diffusive model. Finally, some numerical simulations are carried out to support the analytical results.

Global Bounded Classical Solutions for a Gradient-Driven Mathematical Model of Antiangiogenesis in Tumor Growth

In this paper, we consider a gradient-driven mathematical model of antiangiogenesis in tumor growth. In the model, the movement of endothelial cells is governed by diffusion of themselves and chemotaxis in response to gradients of tumor angiogenic factors and angiostatin. The concentration of tumor angiogenic factors and angiostatin is assumed to diffuse and decay. The resulting system consists of three parabolic partial differential equations. In the present paper, we study the global existence and boundedness of classical solutions of the system under homogeneous Neumann boundary conditions.

Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{x}{l}- $\end{document} like which is unstable in model without nonlocal competition and also greatly different from these with the shape of \begin{document}$ \cos\omega_0 t+\cos\frac{x}{l}- $\end{document} like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of \begin{document}$ \cos\frac{x}{l}- $\end{document} like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{kx}{l}- $\end{document} like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

Existence Results for Fractional Diffusion Equation with Dirichlet Boundary Condition

In this article, we investigate the time-fractional diffusion equation with homogeneous Dirichlet boundary condition. The existence and uniqueness of a solution to the equation is proved using Lax-Milgram theorem. Then the time is discretized with the help of the finite difference scheme and stability result for the semi-discretized problem is shown. Furthermore, space is discretized using the idea of triangulation in the finite element method.

Stability of orthotropic plates

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

Study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

In this paper, we study of the dissipativity, global attractor and exponential attractor for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms, with initial and homogenous Dirichlet boundary condition.

SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.

Wedge and crack solutions for Mindlin–Reissner plates

Displacement potentials are used to define the complete set of displacement and stress-resultant fields in moderately thick plate bending problems. The asymptotic singular terms are used to define and solve the eigenproblems associated with all alternative homogeneous wedge boundary conditions, and extended to model bimaterial wedges. This solution is used to model edge cracks with a filler and thus simulate the effect of crack repair. Definitions for the stress intensity factors are given, as well as the constitutive relations used to model a repaired crack with a contact element.

On time-periodic solutions of some nonlinear parabolic equations with nonmonotone multivalued terms

In this paper we study the existence of time periodic solutions to a class of nonlinear parabolic equations with multivalued nonlinear terms subject to the homogeneous Dirichlet boundary condition. We give two types of existence results: one for large periodic solutions with any large data, and the other for small periodic solutions with small data. Both concern the case where the nonlinear terms contain either a upper semicontinuous multivalued term or a lower semicontinuous multivalued term. Some applications of our results are also given.

Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator

In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value \begin{document}$ \rho $\end{document} of the ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} of diffusions of predator to prey is obtained, such that if \begin{document}$ \frac{d_2}{d_1}>\rho $\end{document} , then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if \begin{document}$ \frac{d_2}{d_1} , both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} . As \begin{document}$ \frac{d_2}{d_1} $\end{document} increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.

Error of the Finite Element Approximation for a Differential Eigenvalue Problem with Nonlinear Dependence on the Spectral Parameter

The positive definite ordinary differential nonlinear eigenvalue problem of the second order with homogeneous Dirichlet boundary condition is considered. The problem is formulated as a symmetric variational eigenvalue problem with nonlinear dependence of the spectral parameter in a real infinite-dimensional Hilbert space. The variational eigenvalue problem consists in finding eigenvalues and corresponding eigenfunctions of the eigenvalue problem for a symmetric positive definite bounded bilinear form with respect to a symmetric positive definite completely continuous bilinear form in a real infinite-dimensional Hilbert space. The variational eigenvalue problem is approximated by the mesh scheme of the finite element method on the uniform grid. For constructing the mesh scheme, Lagrangian finite elements of arbitrary order are applied. Error estimates of approximate eigenvalues and error estimates of approximate eigenfunctions in the norm of initial real infinite-dimensional Hilbert space are established. These error estimates coincide in the order with error estimates of mesh scheme of the finite element method for linear eigenvalue problems. Moreover, superconvergence estimates for approximate eigenfunctions in the mesh norm with Gauss quadrature nodes are derived. Investigations of this paper generalize well known results for the eigenvalue problem with linear entrance on the spectral parameter.

Numerical approximation of the fractional HIV model using the meshless local Petrov\u2013Galerkin method

This paper deals with the model of fractional HIV-1 infection of CD4+T cells transformation with homogeneous Neumann boundary conditions. Numerical methods for solving fractional time differential equations are developed with Caputo’s definition. The forward difference methods were constructed applied to the approximation of the fractional time differential equation. The MLPG method is used to solve the problem of fractional HIV models for spatial discretization. Approximated solutions at the time level n use conventional iterative methods such as fixed point iterations to handle the nonlinear parts. An analysis of stability and convergence of numerical schemes is presented along with the eigenvalue of the matrix. The abilities of the developed formula was confirmed through four numerical examples base on convergence and accuracy of numerical results. The results of the numerical experiments were compared with the solution of the integer order differential equation to confirm the accuracy and efficiency of the proposed scheme. The simulation results show that the formula is easy to use and useful for those interested in fractional derivatives.