Homogeneous Dirichlet Boundary Conditions Research Articles

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when α∈(0,1). We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

Probabilistic analysis of a class of 2D-random heat equations via densities

We give new probabilistic results for a class of random two-dimensional homogeneous heat equations with mixed homogeneous Dirichlet and Neumann boundary conditions and an arbitrary initial condition on a rectangular domain. The diffusion coefficient is assumed to be an arbitrary second-order random variable, while the initial condition is a stochastic process admitting a Karhunen-Loève expansion. We then construct pointwise convergent approximations for the main moments and the density of the solution. The theoretical results are numerically illustrated.

Determining Modes of the 2D g-Navier-Stokes Equations

The “determining modes” introduced by Prodi and Foias in 1967 say that if two solutions agree asymptotically in their P projection, then they are asymptotically in their entirety (see (Foias, 1967)). We study the initial boundary value problem for 2D g-Navier-Stokes (g-NVS) equations in bounded domains with homogeneous Dirichlet boundary conditions. We find an improved upper bound on the number of deterministic modes. Moreover, we slightly improve the estimate of the number of deterministic modes and achieve the upper limit of the Grashof Gr numerical order. These estimates are consistent with heuristic estimates based on physical arguments, extends previous results by O.P. Manley and Y.M. Treve (see (Foias, 1983)). The Gronwall lemma and Poincaré type inequality will play a central role in our computational technique as well as of the paper. Studying the properties of solutions is important to determine the behavior of solutions over a long period of time. The obtained result particularly extends previous results for 2D NVS equations.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Gradient estimates of general nonlinear singular elliptic equations with measure data

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations −divA(x,u,Du)=μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p-Laplacian with respect to Du for the singular case p∈(1,2−1/n], whose discontinuity in the x-variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u-variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good-λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces.

Quantum-information theory of magnetic field influence on circular dots with different boundary conditions

Influence of the transverse uniform magnetic field B on position (subscript ρ) and momentum (γ) Shannon quantum-information entropies S ρ,γ Fisher informations I ρ,γ and informational energies O ρ,γ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Comparative analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. A conspicuous distinction between the two spectra are crossings at the increasing induction of the Neumann energies with the same radial quantum number n and adjacent non-positive angular indices m. At the growing B, either system undergoes Landau condensation when its characteristics turn into their uniform field counterparts. It is shown that for the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g. the Dirichlet sum on its approach from above to a fundamental limit is at any B smaller than the corresponding Neumann quantity what physically means that the former geometry provides more total information about the position and motion of the particle. It is pointed out that the widely accepted disequilibrium uncertainty relation , with d being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed; in particular, contrary to the hard-wall QDs, for this configuration the sums S ρ + S γ and the products I ρ I γ and O ρ O γ do not depend on the field. Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.

Mathematical analysis of an age-since infection and diffusion HIV/AIDS model with treatment adherence and Dirichlet boundary condition

In this paper, an epidemic model with homogeneous Dirichlet boundary condition is formulated to study the joint impact of spatial diffusion, infection age and treatment adherence on the HIV/AIDS transmission among humans. It is an interesting problem to understand the threshold dynamics of HIV/AIDS model with the above three factors. Since the complexity of the model and specificity of the boundary condition, there are two main mathematical challenges: i). the compactness of the solution map is not guaranteed; ii). the explicit expression of the basic reproduction cannot be given even if the parameters are spatially independent. We first discuss the well-posedness of the system, then we identify the basic reproduction number R0 as the spectral radius of the next generation operator, followed by the global attractivity of disease-free steady state when R0<1, the uniform persistence of the disease and the existence of the endemic steady state when R0>1. Numerical simulations are carried out to illustrate our theoretical results, which suggest that the diffusion of individuals has an opposite effect on the disease outbreaks, and improving the treatment compliance of HIV infected individuals can control HIV transmission among the population effectively.

Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source term

We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity utt−Δut−Δpu+ut=|u|q−2uln|u|,x∈Ω,t>0 in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.

Spatiotemporal Patterns of a Host-Generalist Parasitoid Reaction–Diffusion Model

In this paper, we study a delayed host-generalist parasitoid diffusion model subject to homogeneous Dirichlet boundary conditions, where generalist parasitoids are introduced to control the invasion of the hosts. We construct an explicit expression of positive steady-state solution using the implicit function theorem and prove its linear stability. Moreover, by applying feedback time delay [Formula: see text] as the bifurcation parameter, spatially inhomogeneous Hopf bifurcation near the positive steady-state solution is proved when [Formula: see text] is varied through a sequence of critical values. This finding implies that feedback time delay can induce spatially inhomogeneous periodic oscillatory patterns. The direction of spatially inhomogeneous Hopf bifurcation is forward when parameter [Formula: see text] is sufficiently large. We present numerical simulations and solutions to further illustrate our main theoretical results. Numerical simulations show that the period and amplitude of the inhomogeneous periodic solution increase with increasing feedback time delay. Our theoretical analysis results only hold for parameter [Formula: see text] when it is sufficiently close to 1, whereas numerical simulations suggest that spatially inhomogeneous Hopf bifurcation still occurs when [Formula: see text] is larger than 1 but not sufficiently close to 1.

Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

We consider two Gaussian measures μ,μ˜ on a separable Hilbert space, with fractional-order covariance operators A−2β and A˜−2β˜, respectively, and derive necessary and sufficient conditions on A,A˜ and β,β˜>0 for I. equivalence of the measures μ and μ˜, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure μ˜. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and A˜ are elliptic second-order differential operators, formulated on a bounded Euclidean domain D⊂Rd and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.

The hydrodynamic limit for the inhomogeneous Vlasov–Navier–Stokes system

In this paper, we study the hydrodynamic limit for the fluid–particle model, which consists of the inhomogeneous incompressible Navier–Stokes equations coupled with the Vlasov equation through a drag force in a bounded domain of [Formula: see text] with a homogeneous Dirichlet boundary condition on the fluid velocity field and Maxwell boundary condition on the kinetic distribution function. The proof relies on the relative entropy argument, which extends the work of El Ghani [Asymptotic analysis for a Vlasov–Navier–Stokes system in a bounded domain, J. Hyperbolic Differ. Equ. 7 (2010) 191–210] to inhomogeneous incompressible Navier–Stokes–Vlasov equations.

Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis

Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.

A new class of fractional Orlicz-Sobolev space and singular elliptic problems

This paper seeks to introduce a new class of fractional Orlicz-Sobolev space with variable-order. We give basic properties of this space and prove some compactness results. Then, using some techniques of calculus of variations combined with theory of Musielak functions, we prove the existence of a nonnegative weak solution for a singular elliptic type problem in a fractional variable-order Orlicz-Sobolev space with homogeneous Dirichlet boundary conditions.

A Higher-Order Improved Runge–Kutta Method and Cubic B-Spline Approximation for the One-Dimensional Nonlinear RLW Equation

This article developed a significant improvement of a Galerkin-type approximation to the regularized long-wave equation (RLW) solution under homogeneous Dirichlet boundary conditions for achieving higher accuracy in time variables. First, a basis derived from cubic B-splines and limit conditions is used to perform a Galerkin-type approximation. Then, a Crank–Nicolson and fourth-order 4-stage improved Runge–Kutta scheme (IRK4) is used to discretize time. Both a strong stability analysis of a fully discrete IRK4 scheme and the evaluation of Von Neumann stability of the proposed Crank–Nicolson technique are examined. We demonstrate the efficiency of our method with two test problems. The analytical and numerical solutions found in the literature are then contrasted with the approximate solutions produced by the suggested method. The validated numerical results illustrate that the provided technique is more efficient and converges faster than earlier research, resulting in less computational time, smaller space dimensions, and storage. As a result, the proposed numerical approach is appealing for approximating PDEs whose explicit solution is unknown for a variety of boundary conditions.

Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation

In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.

Coexistence solutions for a Lotka–Volterra competition model with density-dependent motion

This paper is concerned with a Lotka–Volterra model with density-dependent motion. Under homogeneous Dirichlet boundary conditions, which is different from the previous article (Liu and Guo, 2021), the corresponding steady-state problem is discussed. Stability properties of the trivial and semitrivial solutions are determined completely. The sufficient conditions for the existence of coexistence solutions are also obtained by using the theory of fixed point index in positive cones. Finally, the limiting behavior of coexistence solutions is studied as some certain parameter tends to infinity.

Weighted mixed-norm Lp estimates for equations in non-divergence form with singular coefficients: The Dirichlet problem

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the Ap-Muckenhoupt class.

Multiple solutions for eigenvalue problems involving the (p,q)-Laplacian

"This paper is devoted to a subject that Professor Csaba Varga suggested during his frequent visits to the University of Perugia and in my regular stays at the ""Babe s-Bolyai"" University. More speci cally, continuing the work started in [7] jointly with Professor Varga, here we establish the existence of two nontrivial (weak) solutions of some one parameter eigenvalue (p,q)-Laplacian problems under homogeneous Dirichlet boundary conditions in bounded domains of R^N."

Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $\varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $\varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.

Periodic solutions for the Blackstock–Crighton–Westervelt equation

We investigate the Blackstock–Crighton–Westervelt equation which models nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We prove existence and regularity, in a Lp−Lq setting, of time-periodic solutions for a given sufficiently small time-periodic forcing data, and homogeneous Dirichlet boundary conditions over a cylindrical domain. We show maximal Lp-regularity for the abstract linearized model. We use techniques of operator-valued Fourier multiplier theorems combined with a generalized version of the implicit function theorem.