Exponential Decay Property Research Articles

Analysis of large urn models with local mean-field interactions

The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time all the balls of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results. When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.

Boundary Conditions and Subelliptic Estimates for Geometric Kramers-Fokker-Planck Operators on Manifolds with Boundaries

This article is concerned with maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic Laplacian.

Asymptotic Analysis and Optimal Error estimates for Benjamin‐Bona‐Mahony‐Burgers' Type Equations

In this article, stabilization result for the Benjamin‐Bona‐Mahony‐Burgers' (BBM‐B) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Based on appropriate conditions on the forcing function, exponential decay estimates in , and ‐norms are derived, which are valid uniformly with respect to the coefficient of dispersion as it tends to zero. It is, further, observed that the decay rate for the BBM‐B equation is smaller than that of the decay rate for the Burgers equation. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and stabilization results are discussed for the semidiscrete problem. Moreover, optimal error estimates in ‐norms preserving exponential decay property are established using the steady state error estimates. For a complete discrete scheme, a backward Euler method is applied for the time discretization and stabilization results are again proved for the fully discrete problem. Subsequently, numerical experiments are conducted, which verify our theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.

On Kirchhoff's Model of Parabolic Type

ABSTRACTIn this article, the existence of a global strong solution for all finite time is derived for the Kirchhoff's model of parabolic type. Based on exponential weight function, some new regularity results which reflect the exponential decay property are obtained for the exact solution. For the related dynamics, the existence of a global attractor is shown to hold for the problem when the non-homogeneous forcing function is either independent of time or in L∞(L2). With the finite element Galerkin method applied in spatial direction keeping time variable continuous, a semidiscrete scheme is analyzed, and it is also established that the semidiscrete system has a global discrete attractor. Optimal error estimates in L∞(H1) norm are derived which are valid uniformly in time. Further, based on a backward Euler method, a completely discrete scheme is analyzed and error estimates are derived. It is also further, observed that in cases where f = 0 or f = O(e−γ0t) with γ0 > 0, the discrete solutions and error estimates decay exponentially in time. Finally, some numerical experiments are discussed which confirm our theoretical findings.

Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both and ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016

Exponential Decay Property for Eigenfunctions of Landau–Stark Hamiltonian

In this paper, we study the exponential decay property for eigenfuctions of a perturbed Landau–Stark Hamiltonian. As a result, we prove this property for long-range Coulomb-type perturbations.

Intrinsic ultracontractivity via capacitary width

The semigroup associated with the Dirichlet heat kernel is said to be intrinsic ultracontractive if (i) the Dirichlet realization of the associated self adjoint operator has the first positive eigenvalue with positive L2 eigenfunction; (ii) the heat kernel is bounded above and below by the product of the eigenfunctions with positive multiplicative constants depending on time. We give an upper and lower estimate of the first eigenvalue in terms of capacitary width, which yields a sharp sufficient condition for (i). Our parabolic argument also yields an exponential decay property of a certain caloric measure. Then, the caloric measure is controlled by the elliptic Green function with the aid of a parabolic box argument. This is a crucial step for (ii). We give a sharp sufficient integral condition for intrinsic ultracontractivity in terms of capacitary width. A similar integral condition for the boundary Harnack principle is also obtained. Under geometric specifications, these integral conditions generalize known results and give more precise conditions. Sharpness is examined by an infinite funnel, for which we obtain a complete characterization of intrinsic ultracontractivity. Our method is purely analytic and elementary; it enables us to dispense with logarithmic Sobolev inequalities.

Scalar field equation with non-local diffusion

In this paper we are interested on the existence of ground state solutions for fractional field equations of the form $$\left\{\begin{array}{ll} (I - \Delta)^{\alpha}u = f(x, u) & \quad {\rm in} \, u > 0 & \quad {\rm in} \, I\!R^N, \quad \displaystyle\lim_{|x| \to \infty}u(x) = 0,\end{array}\right.$$ where \({\alpha \in (0,1)}\) and f is an appropriate super-linear sub-critical nonlinearity. We prove regularity, exponential decay and symmetry properties for these solutions. We also prove the existence of infinitely many bound states and, through a non-local Pohozaev identity, we prove nonexistence results in the supercritical case.

A contact problem for a thermoelastic Timoshenko beam

In this paper, a dynamic contact problem between a Timoshenko beam and two rigid obstacles is considered. Thermal effects are also taken into account and the contact is modeled using the classical Signorini condition. The global existence in time of solutions is found by considering related penalized problems, proving some a priori estimates and passing to the limit. An exponential decay property is also showed.

On the persistence and unique continuation properties for an integrable two-component Dullin–Gottwald–Holm system

Considered herein is the persistence and unique continuation properties of the solution of an integrable two-component Dullin–Gottwald–Holm (DGH2) system, which was derived from the Euler equation with constant vorticity in shallow water waves moving over a linear shear flow. It is proved that the solutions of the DGH2 system enjoy the same exponential decay property as the initial data, as a consequence, if the solutions and their spacial derivatives decay exponentially initially and at a later time, then they must identically equal to zero.

Analysis of dynamic nonlinear thermoviscoelastic beam problems

In this paper, we study a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles. Thermal effects are also taken into account and the contact is modeled using the classical Signorini condition. The existence of solutions is proved by considering approximate problems from a penalization procedure, proving some a priori estimates and passing to the limit. An exponential decay property is also obtained. Then, fully discrete approximations of the approximate problem are provided using the finite element method for the spatial approximation and the implicit Euler scheme for the discretization of the time derivatives. A stability property is obtained and a priori error estimates are proved from which, under some additional regularity conditions, the linear convergence of the algorithm is deduced. Fully discrete approximations of the Signorini problem are then introduced proceeding in a similar way, for which a stability property is obtained. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution.

Stability properties of an abstract system with applications to linear thermoelastic plates

This paper deals with an abstract version of the evolution system $$\left\{\begin{array}{ll}u_{tt}-\omega \Delta u_{tt}+\Delta (\Delta u + \theta)=0 \\\theta_t - \Delta \theta -\Delta u_t=0\end{array}\right.$$ with ω ≥ 0, ruling the dynamics of linear thermoelastic plates. In particular, we will consider a more general coupling term depending on a real parameter σ. The (exponential and polynomial) decay properties of the related solution semigroup are investigated in dependence of the values of σ.

Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

This paper is devoted to the study of a system of nonlinear equations with nonlinear boundary conditions. First, on the basis of the Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, the exponential decay property of the global solution via the construction of a suitable Lyapunov functional is presented. Copyright © 2013 John Wiley & Sons, Ltd.

Gamma-expansion for a 1D confined Lennard-Jones model with point defect

We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the {\Gamma}-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a {\Gamma}-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.

Semidiscrete Galerkin method for equations of motion arising in Kelvin‐Voigt model of viscoelastic fluid flow

AbstractFinite element Galerkin method is applied to equations of motion arising in the Kelvin–Voigt model of viscoelastic fluids for spatial discretization. Some new a priori bounds which reflect the exponential decay property are obtained for the exact solution. For optimal L∞(L2) estimate in the velocity, a new auxiliary operator which is based on a modification of the Stokes operator is introduced and analyzed. Finally, optimal error bounds for the velocity in L∞(L2) as well as in L∞(H)‐norms and the pressure in L∞(L2)‐norm are derived which again preserves the exponential decay property. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

On the Existence of Solitary Traveling Waves for Generalized Hertzian Chains

We consider the question of existence of "bell-shaped" (i.e. non-increasing for x>0 and non-decreasing for x<0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p=3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego [Proceedings of the AMS 133, 1763 (2005)] formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke-Wattis result [Comm. Math. Phys 161, 394 (1994)], by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.

Boundary Stabilization of a More General Kirchhoff-Type Beam Equation

Simultaneously, considering the viscous effect of material, damping of medium, geometrical nonlinearity, physical nonlinearity, we set up a more general equation of beam subjected to axial force and external load. We prove the existence and uniqueness of global solutions under non-linear boundary conditions which the model is added one damping mechanism at l end. What is more, we also prove the exponential decay property of the energy of above mentioned system.

On the Persistence of Decay Properties for the b−Family of Equations

Abstract The b−family of equations governs evolution of shallow water waves with the horizontal velocities in different depths related to the parameter b. Using the method of characteristics, we obtain the results of the exponential decay properties of the strong solutions for the b−family of equations. In particular, we show that a non-trivial strong solution with compact initial value can not be compactly supported at any later time.

Exponential decay and Fredholm properties in second-order quasilinear elliptic systems

We consider second-order quasilinear elliptic systems on unbounded domains in the setting of Sobolev spaces. We complete our earlier work on the Fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be Fredholm of index zero. This opens the way to using the degree for C 1 -Fredholm maps of index zero as a tool in the study of such quasilinear systems. Our work also enables us to check the Fredholm assumption which plays an important role in Rabier's approach to proving exponential decay to zero at infinity of solutions.

Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term

We give an explicit formula for exponential decay properties of positive solutions for a class of semilinear elliptic equations with Hardy term in the whole space Rn.