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In this paper, we establish some new functional inequalities of Adams’ type for radial Sobolev spaces of second order with logarithmic weights. The sharpness of these inequalities is also discussed. These inequalities (which can be singular) are new and complete the existing literature concerning this topic of research. Finally, we will end this work with a concrete application of these new inequalities to study some singular elliptic nonlinear equations involving new type of exponential growth condition at infinity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space R n . Since the standard Sobolev embeddings on R n are not compact and the spectrum of the Laplace operator on R n are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on R n is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces. [ABSTRACT FROM AUTHOR]

We consider in an infinite horizontal layer the stationary motion of a viscous compressible fluid in a magnetic field subject to the gravitational force, where the Dirichlet boundary condition for the velocity and similar but non-homogeneous and large enough conditions for the magnetic field are assumed. Existence of a stationary solution in a neighborhood close to the equilibrium state is obtained in Sobolev spaces as limit of a sequence of fixed points of some suitable operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long term dynamics of the supercritical stochastic wave equation driven by multiplicative noise on unbounded domains. By introducing an appropriate variable, we convert the stochastic equation into a pathwise deterministic wave equation and prove the existence and uniqueness of solutions by the uniform Strichartz estimates. Then we prove the existence and uniqueness of pullback random attractors of the non-autonomous random dynamical system associated with the stochastic equation. The pullback asymptotic compactness of solutions is established by the idea of uniform tail-ends estimates as well as the decomposition method of the solution operator in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic reaction–diffusion equations with superlinear drift on the real line R driven by space–time white noise. A Freidlin–Wentzell large deviation principle is established by a modified weak convergence method on the space C ([ 0 , T ] , C tem (R)) , where C tem (R) : = { f ∈ C (R) : sup x ∈ R | f (x) | e - λ | x | < ∞ for any λ > 0 } . Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space–time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the specially designed family of norms on the Fréchet space C ([ 0 , T ] , C tem (R)) , one-order moment estimates of the stochastic convolution, and two nonlinear Gronwall-type inequalities play an important role. [ABSTRACT FROM AUTHOR]

Control properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in (0,+∞)$(0,+\infty)$. Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator. [ABSTRACT FROM AUTHOR]

In this article, we investigate the dynamics of a nonautonomous stochastic strongly damped wave equation defined on ℝN$$ {\mathrm{\mathbb{R}}}&#x0005E;N $$. We first use the energy equation and tail‐estimates to prove the asymptotic compactness of the solutions and obtain the existence of a unique pullback random attractor for the equation with critical nonlinearity. Then, we give an upper bound of fractal dimension of the random attractor when the nonlinearity is of subcritical growth. The unboundedness of the physical space will impose difficulties on the estimation for the upper bound of fractal dimension of the random attractor. [ABSTRACT FROM AUTHOR]

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections. [ABSTRACT FROM AUTHOR]

The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are stated in terms of the generalized -function. Unlike the author's previous papers, there is no sign condition on the lower-order term of the equation. The existence of a renormalized solution of the Dirichlet problem for this equation is proved in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain. [ABSTRACT FROM AUTHOR]

The dynamics of solutions for the plate equation without delay effects has been investigated by many authors. But there are few works on plate equation with variable delay on unbounded domain. Therefore, we firstly consider the existence of pullback attractor for the plate equation with variable delay on by using the theory of multivalued dynamical system. [ABSTRACT FROM AUTHOR]

For the equation considered in some unbounded mixed domain, uniqueness and existence theorems are proved for a solution to the problem with the missing shift condition on the boundary characteristics and an analogue of the Frankl-type condition on the interval of degeneracy of the equation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast high order method for the time-fractional diffusion equation on unbounded spatial domains. The proposed numerical method is a combination of a time-stepping scheme and spectral method for the spatial discretization. First, we reformulate the unbounded domain problem into a bounded domain problem by introducing suitable artificial boundary conditions. Then the time fractional derivatives involved in the equation and the artificial boundary condition are discretized using the so-called L2 formula and sum-of-exponentials (SOE) approximation. The former has been a popular formula for discretization of the Caputo fractional derivative, while the latter is a computational cost reducing technique frequently employed in recent years for convolution integrals. The spatial discretization makes use of the standard Legendre spectral method. The stability and the accuracy of the full discrete problem are analyzed. Our obtained theoretical results include a rigorous proof of the convergence order for both uniform mesh and graded mesh, and a stability proof for the uniform mesh. Finally, several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

In this paper, we deal with an inverse problem of determining the source function in a kinetic equation that is considered in an unbounded domain with Cauchy data. We prove the uniqueness of the solution of an inverse problem by means of a pointwise Carleman estimate. In recent years, kinetic equations have occurred in a variety of important fields and applications, such as aerospace engineering, semi-conductor technology, nuclear engineering, chemotaxis, and immunology.

In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.

A novel adaptive spectral method has been recently developed to numerically solve partial differential equations (PDEs) in unbounded domains. To achieve accuracy and improve efficiency, the method relies on the dynamic adjustment of three key tunable parameters: the scaling factor, a displacement of the basis functions, and the spectral expansion order. In this paper, we perform the first numerical analysis of the adaptive spectral method using generalized Hermite functions in both one- and multi-dimensional problems. Our analysis reveals why adaptive spectral methods work well when a “frequency indicator” of the numerical solution is controlled. We then investigate how the implementation of the adaptive spectral methods affects numerical results, thereby providing guidelines for the proper tuning of parameters. Finally, we further improve performance by extending the adaptive methods to allow bidirectional basis function translation, and the prospect of carrying out similar numerical analysis to solving PDEs arising from realistic difficult-to-solve unbounded models with adaptive spectral methods is also briefly discussed.

This paper is devoted to the well-posedness of a novel nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results and, moreover, establish a stability result for extended real-valued HVIs with respect to the extended real-valued function as a parameter. Based on the latter general stability result, we provide various stability results for the interface problem, as well as the stability of a related bilateral obstacle interface problem with respect to the obstacles. [ABSTRACT FROM AUTHOR]

In this paper, we deal with an inverse problem of determining the source function in a kinetic equation that is considered in an unbounded domain with Cauchy data. We prove the uniqueness of the solution of an inverse problem by means of a pointwise Carleman estimate. In recent years, kinetic equations have occurred in a variety of important fields and applications, such as aerospace engineering, semi-conductor technology, nuclear engineering, chemotaxis, and immunology. [ABSTRACT FROM AUTHOR]

The global dynamics of the typical age-structured model with age-dependent mortality and diffusion rates on unbounded domains have been established. On the one hand, we showed that a positive and constant state solution of the mature population is globally asymptotically stable with respect to the compact-open topology; on the other hand, we showed that the trivial solution is globally asymptotically stable with respect to the usual supremum norm. As an application of our result, we applied the result to birth functions appearing in biology. In addition to the theoretical results, we also present a numerical simulation. [ABSTRACT FROM AUTHOR]

This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional p-Laplace operator in ℝ N {\mathbb{R}^{N}} . Precisely, we study the Kirchhoff-type problem ( a + b ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ d x ⁢ d y ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u )   in  ⁢ ℝ N , \Biggl{(}a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,% \mathrm{d}x\,\mathrm{d}y\Biggr{)}(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u)\quad% \text{in }\mathbb{R}^{N}, where a , b > 0 {a,b>0} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian with 0 < s < 1 < p < N s {0V can have negative values and f fulfills suitable growth assumptions. According to the interaction between the attenuation of the potential at infinity and the behavior of the nonlinear term at the origin, using a penalization argument along with L ∞ {L^{\infty}} -estimates and variational methods, we prove the existence of a positive solution. In addition, we also establish the existence of infinitely many solutions provided the nonlinear term is odd. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior of nonclassical diffusion equations with time‐dependent perturbed parameter and lacking instantaneous damping is considered on whole space ℝn$$ {\mathrm{\mathbb{R}}}^n $$. The existence and regularity of time‐dependent global attractors are proved by using a new analytical technique. It is remarkable that there is not any restriction on the upper growth of the nonlinearity f$$ f $$. The result obtained essentially extends and improves earlier ones. [ABSTRACT FROM AUTHOR]

In this work, a stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions (ABCs) is designed for solving a peridynamic diffusion model on the whole two dimensional domain. To do so, we first use a high-order quadrature-based finite difference scheme to approximate the spatially nonlocal operator and use the BDF2 scheme to approximate the temporal direction. For the resulting fully discrete system, we apply the nonlocal potential theory to obtain the discrete Dirichlet-to-Dirichlet (DtD)-type ABCs. After that, we further derive the Dirichlet-to-Neumann (DtN)-type ABCs based on the nonlocal Neumann data defined from the discrete nonlocal Green's first identity. For the stability and convergence analysis, we reformulate the BDF2 operator into a discrete convolution sum form, then use the technique of the discrete orthogonal convolution kernels to obtain the optimal convergence order. Finally, numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given. [ABSTRACT FROM AUTHOR]

In various applications, the adoption of optimal energy matching pursuit with dictionary elements is common. When the dictionary elements are indexed by parameters within a bounded region, exhaustion-type algorithms can be employed. This article aims to investigate a process that converts the optimal parameter selection in unbounded regions to a bounded and closed (compact) sub-domain. Such a process provides accessibility for energy matching pursuit in a wide range of applications. The paper initially focuses on the open unit disc and the upper-half complex plane, introducing adaptive Fourier decomposition as the underlying methodology. It then extends this concept to general Hilbert spaces with a dictionary and Bochner spaces for random signals. Computational examples are included to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior of nonclassical diffusion equations with hereditary memory and time-dependent perturbed parameter on whole space $ \mathbb{R}^n $. Under a general assumption on the memory kernel $ k $, the existence and regularity of time-dependent global attractors are proven using a new analytical technique. It is remarkable that the nonlinearity $ f $ has no restriction on the upper growth.

• A novel time-domain 3D artificial boundary for poroelastic medium. • Capability to model boundaries with general geometries. • Capability to consider planar physical interfaces extending to infinity. • Seamless coupling with poroelastic finite-element codes based on Biot's theory. In this paper, we introduce a novel direct time-domain artificial boundary condition, called the scaled boundary perfectly matched layer, for wave problems in 3D unbounded fluid-saturated poroelastic medium. The poroelastic medium is conceptualized as a two-phase medium composed of a solid skeleton and an interstitial fluid based on the u- p formulation of Biot's theory. The scaled boundary perfectly matched layer proposed in this work originates from an extension of a recently developed perfectly matched layer for wave problems in a single-phase solid medium. The dependence of the conventional perfectly matched layer on the global coordinate system is overcome in the present method. As a result, this method permits the utilization of an artificial boundary with general geometry (not necessarily convex) and can consider planar physical surfaces and interfaces extending to infinity. The proposed method is formulated by a mixed unsplit-field form, which includes both the mixed displacement-stress field and pore pressure-relative displacement field. The resulting scaled boundary perfectly matched layer formulation can be directly described by a system of third-order ordinary differential equations in time, which can be easily integrated into the standard finite-element framework of poroelastic theory. Finally, numerical benchmark examples are presented to demonstrate the accuracy and robustness of the proposed scaled boundary perfectly matched layer, as well as the versatility in practical engineering problems. [ABSTRACT FROM AUTHOR]

We consider a second-order quasilinear elliptic equation with an integrable right-hand side. We formulate constraints on the structure of the equation in terms of a generalized -function. We prove the existence of an entropic solution of the Dirichlet problem in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain. [ABSTRACT FROM AUTHOR]

In this work, we prove the existence of a nontrivial solution to a quasilinear elliptic problem defined on the whole Euclidean space R N , N ≥ 2 , and involving a weighted 1-Laplacian operator. The nonlinear term has a singular behavior at the origin. This solution is obtained through an approximation technique, which consists in considering the problem with the 1-Laplacian operator as a limit of a family of problems with the p-Laplacian operators when p → 1 +. For that aim, a new version of Anzellotti’s L ∞ - divergence−measure pairing theory is established and new arguments are used. [ABSTRACT FROM AUTHOR]

Lattice Green's functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2-dimensional or 3-dimensional (3D) LGFs in linear time, avoiding the need for brute-force multidimensional quadrature of numerically unstable integrals. Here we focus on higherorder discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with highorder dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson's equation on fully or partially unbounded 3D domains. [ABSTRACT FROM AUTHOR]

We study the first initial boundary value problem for the non-autonomous 2D g-Navier–Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. We show the existence and some further results of a pullback attractor for the process generated by strong solutions to the problem with respect to a large class of non-autonomous forcing terms. To overcome the difficulty caused by the unboundedness of the domain, the proof is based on a pullback asymptotic compactness argument and the use of the enstrophy equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of the non-local fractional stochastic reaction-diffusion equation with a polynomial drift of arbitrary degree driven by multiplicative noise defined on unbounded domains. We first prove the strong convergence of the solutions of a control equation with respect to the weak topology of controls, and then show the convergence in distribution of the solutions of the stochastic equation when the noise intensity approaches zero. We finally establish the large deviations of the stochastic equation by the weak convergence method. The main difficulty of the paper is caused by the non-compactness of Sobolev embeddings on unbounded domains, and the idea of uniform tail-ends estimates is employed to circumvent the obstacle in order to obtain the tightness of distribution laws of the stochastic equation and the precompactness of the control equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give an existence theorem of a solution of the nonlinear evolution dam problem (P)$$ (P) $$, associated with compressible or incompressible flows, in an unbounded domain Q∞$$ {Q}_{\infty } $$ of ℝn+1$$ {\mathrm{\mathbb{R}}}&amp;#x0005E;{n&amp;#x0002B;1} $$. We approximate Q∞$$ {Q}_{\infty } $$ by a sequence of bounded domains (Qr)r$$ {\left({Q}_r\right)}_r $$, and we consider a regularized problem (Prϵ)$$ \left({P}_{r\epsilon}\right) $$ on the bounded domain Qr$$ {Q}_r $$ which has a unique solution. Now, we pass successively to the limit in (Prϵ)$$ \left({P}_{r\epsilon}\right) $$, as r→∞$$ r\to \infty $$ and then as ϵ→0$$ \epsilon \to 0 $$ to get a solution to (P)$$ (P) $$. Note that in a previos study of Gilardi and Kröner [7], an existence of a solution was given by using a different technique for the evolution dam problem related to an incompressible flow through a homogeneous porous medium. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous non-local fractional stochastic p-Laplacian equations with delay driven by linear multiplicative white noise on the entire space R n . We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

Let F: ℝn → [0, + ∞) be a convex function of class C2(ℝn{0}) which is even and positively homogeneous of degree 1, and its polar F0 represents a Finsler metric on ℝn. The anisotropic Sobolev norm in W1,n(ℝn) is defined by ‖ u ‖ F = ( ∫ ℝ n (F n (∇ u) + | u | n) d x ) 1 / n. In this paper, the following sharp anisotropic Moser–Trudinger inequality involving Ln norm sup u ∈ W 1 , n (ℝ n) , ‖ u ‖ F ≤ 1 ∫ ℝ n Φ (λ n | u | n n − 1 (1 + α ‖ u ‖ n n) 1 n − 1 ) d x < + ∞ in the entire space ℝn for any 0 < α < 1 is established, where Φ (t) = e t − ∑ j = 0 n − 2 t j j ! , λ n = n n n − 1 κ n 1 n − 1 and κn is the volume of the unit Wulff ball in ℝn. It is also shown that the above supremum is infinity for all α ≥ 1. Moreover, we prove the supremum is attained, that is, there exists a maximizer for the above supremum when α > 0 is sufficiently small. [ABSTRACT FROM AUTHOR]

In this work, in an unbounded domain, we prove the correctness of the problem with combined Tricomi and Frankl conditions on one boundary characteristic for one class of mixed-type equations. [ABSTRACT FROM AUTHOR]

Let F: ℝn → [0, + ∞) be a convex function of class C2(ℝn{0}) which is even and positively homogeneous of degree 1, and its polar F0 represents a Finsler metric on ℝn. The anisotropic Sobolev norm in W1,n(ℝn) is defined by ‖ u ‖ F = ( ∫ ℝ n (F n (∇ u) + | u | n) d x ) 1 / n. In this paper, the following sharp anisotropic Moser–Trudinger inequality involving Ln norm sup u ∈ W 1 , n (ℝ n) , ‖ u ‖ F ≤ 1 ∫ ℝ n Φ (λ n | u | n n − 1 (1 + α ‖ u ‖ n n) 1 n − 1 ) d x < + ∞ in the entire space ℝn for any 0 < α < 1 is established, where Φ (t) = e t − ∑ j = 0 n − 2 t j j ! , λ n = n n n − 1 κ n 1 n − 1 and κn is the volume of the unit Wulff ball in ℝn. It is also shown that the above supremum is infinity for all α ≥ 1. Moreover, we prove the supremum is attained, that is, there exists a maximizer for the above supremum when α > 0 is sufficiently small. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic compactness of the solutions of the two-dimensional Navier-Stokes equations defined in unbounded channel-like domains. In order to overcome the non-compactness of Sobolev embeddings in unbounded domains, we establish the uniform tail-ends estimates of solutions by showing all the solutions are uniformly small outside a sufficiently large bounded domain for all large time and bounded initial data, which implies the asymptotic compactness and hence the existence of global attractors in the natural energy space. The uniform tail-ends estimates are derived by the scalar stream function of the Navier-Stokes equations which is distinct from the reaction-diffusion equation due to the divergence free condition. [ABSTRACT FROM AUTHOR]

Summary: Flow calculations in an unbounded domain have limitations and challenges due to its infiniteness. A common approach is to impose a far‐field asymptotic condition to determine a unique flow. The leading behavior of the flow is identified at the far field, and then an unknown coefficient is assumed for the second behavior. This allows us to propose an efficient numerical method to solve two‐dimensional steady Stokes and potential flows in a truncated domain along with the coefficients. The second term provides crucial hydrodynamic information for the flow and is referred to as the informative boundary condition. The truncation creates artificial boundaries requiring boundary conditions for the approximate solution. The axial Green function method (AGM), combined with a specific one‐dimensional Green function over a semi‐infinite axis‐parallel line extended to infinity, allows us to implement the informative boundary condition in the truncated domain. AGMs, designed for complicated domains, are now applied to infinite domain cases because AGMs' versatility enables implementing the informative boundary condition by changing only the axial Green function. This approach's efficiency, accuracy, and consistency are investigated through several appealing Stokes flow problems including potential ones in infinite domains. [ABSTRACT FROM AUTHOR]

In this paper, by using the Ge and Ren extension of coincidence degree theory, we established the existence of a solution for a resonant mixed fractional order p-Laplacian boundary value problem (BVP) on the half-line. In the process, we solved the corresponding homogeneous fractional order BVP for conditions critical for resonance and showed that the operator A(x,λ)(t) constructed from the abstract equation Mx(t)=Nx(t) is relatively compact. The results are demonstrated with an example.

In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the $m(x)$-Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere.

We research the well-posedness of the problem without initial condition for nonlinear parabolic variational inequalities with variable time-delay. To justify our results, we impose some assumptions on the solution behavior and growth of the data-in as time variable tends to −∞. Also, we obtain estimates for weak solutions of this problem.

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of [32] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space. [ABSTRACT FROM AUTHOR]

A novel artificial boundary method called the scaled boundary perfectly matched layer (SBPML) is proposed for the transient analysis of elastic wave propagation in unbounded domains. This method constructs a generalized perfectly matched layer (PML) around the finite interior subdomain based on a coordinate transformation inspired by the scaled boundary finite element method (SBFEM). It can model the infinite exterior subdomains of heterogeneous materials, featuring both parallel and radial straight‐line physical surfaces and interfaces. In addition, generally‐shaped (even nonconvex‐shape) artificial boundaries can be used to provide high flexibility in choosing the geometry of the finite interior subdomain. Two coordinate mapping technologies, namely the modified scaled boundary coordinate transformation from the SBFEM and the complex coordinate stretching from the PML, are successively utilized to construct the proposed method. By introducing the integral of stress as an auxiliary variable, the resulting SBPML subdomain is represented in the time domain using a mixed displacement‐stress unsplit‐field formulation. This has the added benefit of allowing for easy incorporation into standard dynamic finite element methods. The performance of the SBPML is demonstrated through several numerical examples, including P‐SV and SH wave propagation problems occurring in unbounded domains with complex geometries and heterogeneous material properties. The numerical results are evaluated in comparison to both reference solution and existing artificial boundary methods, ultimately demonstrating the flexibility, robustness, accuracy, and stability of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a uniform contraction principle for verifying the uniform large deviation principles of locally Hölder continuous maps in Banach spaces. We then show the local Hölder continuity of the solutions of a class of fractional parabolic equations with polynomial drift of any order defined on ℝ n . We finally establish the large deviation principle of the fractional stochastic equations uniformly with respect to bounded initial data, despite the solution operators are not compact due to the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]