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We are concerned with the linear problem - Δ u + κ | x | 2 x · ∇ u = λ K (| x |) u , x ∈ R N , u (x) > 0 , x ∈ R N , [ 2 e x ] u (x) → 0 , | x | → ∞ , where λ is a positive parameter, κ ∈ [ 0 , N - 2) , N > 2 and K : R N → (0 , ∞) is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue λ 1 rad and the corresponding positive eigenfunction φ 1 satisfies lim | x | → ∞ φ 1 (| x |) = c | x | N - 2 - κ for some c > 0 . As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the fractional p-Laplacian equation with colored noise on R n . We first obtain the well-posedness and then prove the existence and uniqueness of tempered pullback random attractors for the fractional p-Laplacian equation with nonlinear diffusion term. Furthermore, we show that the pullback random attractors of the systems driven by linear colored noise converge to that of the corresponding stochastic systems driven by linear white noise under the topology of L 2 as the correlation time of the colored noise approaches zero. [ABSTRACT FROM AUTHOR]

Rapidly developing machine learning methods have stimulated research interest in computationally reconstructing differential equations (DEs) from observational data, providing insight into the underlying mechanistic models. In this paper, we propose a new neural-ODE-based method that spectrally expands the spatial dependence of solutions to learn the spatiotemporal DEs they obey. Our spectral spatiotemporal DE learning method has the advantage of not explicitly relying on spatial discretization (e.g., meshes or grids), thus allowing reconstruction of DEs that may be defined on unbounded spatial domains and that may contain long-ranged, nonlocal spatial interactions. By combining spectral methods with the neural ODE framework, our proposed spectral DE method addresses the inverse-type problem of reconstructing spatiotemporal equations in unbounded domains. Even for bounded domain problems, our spectral approach is as accurate as some of the latest machine learning approaches for learning or numerically solving partial differential equations (PDEs). By developing a spectral framework for reconstructing both PDEs and partial integro-differential equations (PIDEs), we extend dynamical reconstruction approaches to a wider range of problems, including those in unbounded domains. [ABSTRACT FROM AUTHOR]

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space Hl⊂Hs(Rm) (s≥1) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in Hl as well as bounded linear operators from L² to Hl. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in H4(R) as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave. [ABSTRACT FROM AUTHOR]

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like | E (x) | | ∇ u | , where E (x) is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for computing the stray-field and the corresponding energy for a given magnetization configuration. Our approach is based on the use of inverted finite elements and does not need any truncation. After analyzing the problem in an appropriate functional framework, we describe the method and we prove its convergence. We then display some computational results which demonstrate its efficiency and confirm its full potential. [ABSTRACT FROM AUTHOR]

Abstract: Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient numerical methods that can resolve the dependence of the PDE on the unbounded variable over at least several orders of magnitude. We propose a solution to such problems by combining two classes of numerical methods: (i) adaptive spectral methods and (ii) physics-informed neural networks (PINNs). The numerical approach that we develop takes advantage of the ability of PINNs to easily implement high-order numerical schemes to efficiently solve PDEs and extrapolate numerical solutions at any point in space and time. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs in solving PDEs and estimating model parameters from noisy observations in unbounded domains.

Based on the existence of pullback attractors for the non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping defined in the entire space Rn by Xiaobin Yao in [15], in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating mean dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay on unbounded domains. We first prove the existence of a unique solution to stochastic 3D LANS equations with infinite delay when the non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and the nonlinear diffusion term is locally Lipschitz continuous. This enables us to define a mean random dynamical system. Besides, we find that such a dynamical system possesses a unique weak pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting set. Furthermore, we prove the existence and uniqueness of stationary solutions (equilibrium solutions) to the corresponding deterministic equation via the classical Galerkin method, the Lax–Milgram and the Brouwer fixed theorems. The stability properties of stationary solutions are also considered. By a direct approach, we first show the local stability of stationary solutions when the delay term has a general form and then apply the abstract results to two kinds of infinite delays. Second, we establish the exponential stability of stationary solutions in the case of unbounded distributed delay. Third, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we discuss the polynomial asymptotic stability in the particular case of proportional delay. [ABSTRACT FROM AUTHOR]

In this work, we consider the 2D and 3D SCBF equations driven by irregular additive white noise$ \mathrm{d}\boldsymbol{u}-[\mu \Delta\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-\alpha\boldsymbol{u}-\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nabla p]\mathrm{d}t = \boldsymbol{f}\mathrm{d}t+\mathrm{d}\mathrm{W},\ \nabla\cdot\boldsymbol{u} = 0, $for $ r\in[1,\infty), $ $ \mu,\alpha,\beta>0 $ in unbounded domains (like Poincaré domains) $ \mathcal{O}\subset\mathbb{R}^d $ ($ d = 2,3 $), where $ \mathrm{W}(\cdot) $ is a Hilbert space valued Wiener process on a given filtered probability space, and discuss the asymptotic behavior of its solution. For $ d = 2 $ with $ r\in[1,\infty) $ and $ d = 3 $ with $ r\in[3,\infty) $ (for $ d = r = 3 $ with $ 2\beta\mu\geq 1 $), we first prove the existence and uniqueness of a weak solution (in the analytic sense) satisfying the energy equality for SCBF equations by using a Faedo-Galerkin approximation technique. Then, we establish the existence of random attractors for the stochastic flow generated by the SCBF equations. One of the technical difficulties associated with irregular white noise is overcome with the help of the corresponding Reproducing Kernel Hilbert space. Furthermore, we observe that the regularity of irregular white noise needed to obtain random attractors for the SCBF equations for $ d = 2 $ with $ r\in[1,3] $ and $ d = r = 3 $ with $ 2\beta\mu\geq1 $, is the same as that in the case of 2D Navier-Stokes equations, whereas for the cases $ d = 2,3 $ and $ r\in(3,\infty) $, we require more spatial regularity on the noise. Finally, we address the existence of a unique invariant measure for 2D and 3D SCBF equations defined on Poincaré domains (bounded or unbounded). [ABSTRACT FROM AUTHOR]

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on R d . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on R 3 , the Swift–Hohenberg equation on R 2 , and a three-component FitzHugh–Nagumo system on R 2 . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs. [ABSTRACT FROM AUTHOR]

In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz–Sobolev spaces. What matters here is the behavior of the integral, not the space.

When using spectral methods, a consistent method for tuning the expansion order is often required, especially for time-dependent problems in which oscillations emerge in the solution. In this paper, we propose a frequency-dependent p-adaptive technique that adaptively adjusts the expansion order based on a frequency indicator. Using this p-adaptive technique, combined with recently proposed scaling and moving techniques, we are able to devise an adaptive spectral method in unbounded domains that can capture and handle diffusion, advection, and oscillations. As an application, we use this adaptive spectral method to numerically solve Schrödinger's equation in an unbounded domain and successfully capture the solution's oscillatory behavior at infinity.

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like |E(x)||∇u|, where E(x) is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates.

In this manuscript, we consider the internal control problem for the fifth‐order KdV type equation, commonly called the Kawahara equation, on unbounded domains. Precisely, under certain hypotheses over the initial and boundary data, we can prove that there exists an internal control input such that solutions of the Kawahara equation satisfy an integral overdetermination condition. This condition is satisfied when the domain of the Kawahara equation is posed in the real line, left half‐line, and right half‐line. Moreover, we are also able to prove that there exists a minimal time in which the integral overdetermination condition is satisfied. Finally, we show a type of exact controllability associated with the "mass" of the Kawahara equation posed in the half‐line. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of stochastic plate equations with memory in unbounded domains. More specifically, we obtain the uniform in time estimates for solutions of the problem. Based on the estimates above, we prove the existence and uniqueness of random attractors in unbounded domains. Finally, we show the upper semicontinuity of the attractors when stochastic perturbations approaches zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long-time dynamics and asymptotic stability for the 3D Brinkman-Forchheimer equation with delay on some unbounded domains. The global well-posedness of the 3D Brinkman-Forchheimer flow is investigated by a truncation method firstly, then the pullback attractor has been presented for the process generated by weak solutions. Furthermore, the exponential stability between evolutionary and stationary solutions has been shown by some estimate of elliptic equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations.

In this paper, we propose an efficient numerical scheme with linear complexity for the one-dimensional time-dependent Schrödinger equation on unbounded domains. The artificial boundary method is used to address the unboundedness of the domain. By applying the two-step backward difference formula for time discretization and performing the Z -transform, we derive an exact semi-discrete artificial boundary condition of the Dirichlet-to-Neumann type. To expedite the discrete temporal convolution involved in the exact semi-discrete artificial boundary conditions, we design a fast algorithm based on the best relative Chebyshev approximation of the square-root function. The Galerkin finite element method is used for spatial discretization. By introducing a constant damping term to the original Schrödinger equation, we present a complete error estimate for the fully discrete problem. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous works in this direction our theory does not require the underlying domain to be cylindrical or cone-like or to be shift semiinvariant with respect to some direction. In particular, the theory is applicable to the exterior domains. [ABSTRACT FROM AUTHOR]

We study the computability of global solutions to linear Schrödinger equations with magnetic fields and the Hartree equation on ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$. We show that the solution can always be globally computed with error control on the entire space if there exist a priori decay estimates in generalized Sobolev norms on the initial state. Using weighted Sobolev norm estimates, we show that the solution can be computed with uniform computational runtime with respect to initial states and potentials. We finally study applications in optimal control theory and provide numerical examples. [ABSTRACT FROM AUTHOR]

Let be an unbounded open subset of , . We obtain blow-up conditions for non-negative solutions of the problem in , where and are some function and is the -Laplace operator. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity.

This paper is concerned with the asymptotic behavior of the stochastic three dimensional Brinkman-Forchheimer equations in some unbounded domains. We first define a continuous random dynamical system for the equations. Then by J. Ball's idea of energy equations, we obtain pullback asymptotic compactness of solutions and prove that the existence of a unique random attractor for the equations. [ABSTRACT FROM AUTHOR]

A general approach to solving the Dirichlet problem, both for bounded 3 D domains and for their unbounded complements, in terms of the fractional ( 3 D ) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown. [ABSTRACT FROM AUTHOR]

This article presents a new Carleman inequality for a linear Schrödinger equation which is suitable for both bounded and unbounded domains. We characterize the conditions on the auxiliary function necessary to obtain the global inequality. The novelty of this result is the construction of the auxiliary function on some unbounded domains and for a corresponding valid control region ω . As a consequence, we prove some results on the controllability of a linear Schrödinger equation on unbounded domains. [ABSTRACT FROM AUTHOR]

In this article, we consider the (double) minimization problem min ⁡ { P ⁢ (E ; Ω) + λ ⁢ W p ⁢ (E , F) : E ⊆ Ω , F ⊆ R d , | E ∩ F | = 0 , | E | = | F | = 1 } , where λ ⩾ 0 , p ⩾ 1 , Ω is a (possibly unbounded) domain in R d , P ⁢ (E ; Ω) denotes the relative perimeter of 퐸 in Ω and W p denotes the 푝-Wasserstein distance. When Ω is unbounded and d ⩾ 3 , it is an open problem proposed by Buttazzo, Carlier and Laborde in the paper On the Wasserstein distance between mutually singular measures. We prove the existence of minimizers to this problem when the dimension d ⩾ 1 , 1 p + 2 d > 1 , Ω = R d and 휆 is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we study a Dirichlet problem for noncoercive nonlinear elliptic equations with first order term in an unbounded domain. We obtain Stampacchia type existence, regularity and uniqueness results, when the singular drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. The main tools are a weak maximum principle together with some a priori estimates proved by contradiction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of stochastic plate equations with memory in unbounded domains. More specifically, we obtain the uniform time estimates for solutions of the problem. Based on the estimates above, we prove the existence and uniqueness of random attractors in unbounded domains.

This paper aims at developing a dissipation-preserving, linearized, and time-stepping-varying spectral method for strongly damped nonlinear wave system in multidimensional unbounded domains R d (d=1, 2, and 3), where the nonlocal nature is described by the mixed fractional Laplacians. Because the underlying solutions of the problem involving mixed fractional Laplacians decay slowly with certain power law at infinity, we employ the rational spectral-Galerkin method using rational basis (or mapped Gegenbauer functions) for the spatial approximation. To capture the intrinsic dissipative properties of the model equations, we combine the Crank-Nicolson scheme with exponential scalar auxiliary variable approach for the temporal discretization. Based on the rate of nonlocal energy dissipation, we design a novel time-stepping-varying strategy to enhance the efficiency of the scheme. We present the detailed implementation of the scheme, where the main building block of the stiffness matrices is based on the Laguerre-Gauss quadrature rule for the modified Bessel functions of the second kind. The existence, uniqueness, and nonlocal energy dissipation law of the fully discrete scheme are rigourously established. Numerical examples in 3D case are carried out to demonstrate the accuracy and efficiency of the scheme. Finally, we simulate the nonlinear behaviors of 2D/3D dissipative vector solitary waves for damped sine-Gordon system I, for damped sine-Gordon system II, and for damped Klein–Gordon system to provide a deeper understanding of nonlocal physics. [ABSTRACT FROM AUTHOR]

A biharmonic problem with mixed Steklov-type and Neumann conditions on the boundary in the exterior of a compact set is considered under the assumption that the generalized solutions of this problem have a bounded weighted Dirichlet integral. To solve this biharmonic problem, we use the variational principle and, depending on the value of the parameter included in the weighted integral, we obtain uniqueness (non-uniqueness) theorems, or the same as obtaining exact formulas for calculating the dimension of the solution space. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R 2 be an unbounded convex domain. It is proven in López (J Diff Equ 171:54–62, 2001) and Ripoll (Pac J Math 198(1):175–196, 2001) that, given H > 0 , there exists a graph G ⊂ R 3 of constant mean curvature H over Ω with ∂ G = ∂ Ω if and only if Ω is included in a strip of width 1/H. In this paper we obtain results in H 2 × R in the same direction: given H ∈ 0 , 1 / 2 , if Ω is included in a region of H 2 × 0 bounded by two hypercircles equidistant ℓ (H) to a same geodesic, we show that, if the geodesic curvature of ∂ Ω is bounded from below by - 1 , then there is an H-graph G over Ω with ∂ G = ∂ Ω . We also present more refined existence results involving the curvature of ∂ Ω , which can also be less than - 1. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity. [ABSTRACT FROM AUTHOR]

We discuss some attempts to apply the classical variational methods, in the spirit of Ambrosetti–Rabinowitz, to parametrized elliptic problems defined on unbounded domains. We construct some min-max classes and establish some estimates which allow, for instance, to conclude that, while for a generic coefficient there is always a solution for all small values of the parameter, in the case of a coefficient with a suitable exponential decay and in dimension N D 2 the set of the values of the parameter for which the problem has a nontrivial solution is unbounded. [ABSTRACT FROM AUTHOR]

In this article, we prove an existence result and some regularity for the solution of the strongly nonlinear p(x)-elliptic problem of the form: - div a (x , u , ∇ u) + c (x , u) + H (x , u , ∇ u) = f (x) - div g (x) in Ω , u ∈ W 0 1 , p (.) (Ω) , where H (x , s , ξ) and c(x, s) are a nonlinear terms satisfying some growth condition but no sign condition on H. The assumptions on the source terms load to the existence of solutions. The domain Ω is allowed to have infinite Lebesgue measure. [ABSTRACT FROM AUTHOR]

In this article, we construct an efficient numerical scheme to solve the coupled nonlinear Klein–Gordon system in a multi-dimensional unbounded domain R d (d = 1 , 2 , 3). We first use the SAV method to equivalently deform the original system, then we treat the nonlinear part of the system explicitly in temporal discretization. Thanks to the SAV method, we ensure the unconditional energy conservation of the discrete system, as a cost, we only need to solve two completely linearized decoupled systems. To avoid the errors caused by domain truncation and the construction of artificial boundary conditions, we apply the Hermite–Galerkin spectral method with scaling factor for spatial discretization. In order to save computational costs, we utilize two time adaptive strategies to adjust the time step size, through which we can improve computational efficiency without sacrificing accuracy. Finally, we presented numerical experiments to demonstrate the accuracy, efficiency, and energy conservation properties of our algorithm, and applied it to simulate the interactions of two-dimensional and three-dimensional vector solitons. [ABSTRACT FROM AUTHOR]

This work describes a mesh-free collocation formulation for the acoustic fluid–fluid coupling among non-homogeneous bounded and unbounded fluid domains. The proposed formulations use series of functions that are generated using the analytical solutions of Helmholtz equation in spherical coordinates, and coefficients of the series are calculated from applying the boundary conditions. The technique is able to calculate the acoustic pressure in any bounded or unbounded non-spherical domain. The accuracy, efficiency and robustness of the model is tested by using as benchmark classical analytic solutions of radiation and scattering from spheres.

In this paper, we study the asymptotic behavior of the non-autonomous stochastic 3D Brinkman-Forchheimer equations on unbounded domains. We first define a continuous non-autonomous cocycle for the stochastic equations, and then prove that the existence of tempered random attractors by Ball's idea of energy equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time. [ABSTRACT FROM AUTHOR]

This paper presents an efficient hybrid asynchronous three-dimensional (3D) perfectly matched layer (PML) for modeling unbounded domains. The proposed unsplit explicit or implicit 3D PML formulation is implemented in the framework of a heterogeneous asynchronous time integrator. It is fully versatile in terms of time integrators and time step sizes according to partitions while conserving classical finite element formulations in the elastic domain without complex-valued stretched coordinates. Examples of a semi-infinite bar, Lamb’s test, and a soil–structure interaction problem with PML-truncated semi-infinite heterogeneous media are investigated to illustrate the efficiency of the proposed PML in terms of accuracy and CPU time.

We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in R 3 R 3 are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method. [ABSTRACT FROM AUTHOR]

Abstract Based on the abstract theory of pullback attractors of non-autonomous non-compact dynamical systems by differential equations with both dependent-time deterministic and stochastic forcing terms, introduced by Wang in (J. Differ. Equ. 253:1544–1583, 2012), we investigate the existence of pullback attractors for the non-autonomous stochastic plate equations with additive noise and nonlinear damping on R n $\mathbb{R}^{n}$ .