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Ordinary and partial differential equations with operators containing the p -Laplace operator, the so called ϕ -Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered. [ABSTRACT FROM AUTHOR]

In this paper we ascertain the singular point-wise behavior of the positive solutions of a semilinear elliptic boundary value problem (1) at the critical value of the parameter, λ , where it begins its metasolution regime. As the weight function m (x) changes sign in Ω, our result is a substantial extension of a previous, very recent, result of Li et al. [8] , where it was imposed the (very strong) condition that m ≥ 0 on a neighborhood of b − 1 ({ 0 }). In this paper, we are simply assuming that m (x 0) > 0 for some x 0 ∈ b − 1 ({ 0 }). • Theorem 1.1 proves that the behavior of the solutions proved by Li et al. [8] also occurs with much weaker hypotheses. • Theorem 3.1 is a substantial extension of Theorem 2.1 of López-Gómez and Sabina de Lis [12]. • Lemma 2.1 provides a useful estimate of eigenfunctions associated to an eigenvalue problem with sign changing weight. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following boundary value problem { − Δ p u + g (u) | ∇ u | p = h (u) f in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, in a domain Ω ⊂ R N (N ≥ 2) , where 1 ≤ p < N , g is a positive and continuous function on [ 0 , ∞) , and h is a continuous function on [ 0 , ∞) (possibly blowing up at the origin). We show how the presence of regularizing terms h and g allows to prove existence of finite energy solutions for nonnegative data f only belonging to L 1 (Ω). [ABSTRACT FROM AUTHOR]

In this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. The solution operator of the quasiperiodic boundary value problem yields a well-defined scattering operator, the numerical approximation of which is nothing else but the S-matrix in scattering matrix algorithms like Rigorous Coupled Wave Analysis (RCWA) or Fourier Modal Method (FMM). [ABSTRACT FROM AUTHOR]

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method. [ABSTRACT FROM AUTHOR]

The Hamiltonian systems considered in this paper are obtained by weakly coupling two systems having completely different behaviors. The first one satisfies the twist assumptions usually considered for the application of the Poincaré–Birkhoff Theorem, while the second one presents the existence of some well-ordered lower and upper solutions. In the higher dimensional case, we also treat a coupling situation where the classical Hartman condition is assumed. [ABSTRACT FROM AUTHOR]

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem. [ABSTRACT FROM AUTHOR]

Abstract: In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum forces solutions to have higher integrability as well. In the present paper, we consider -obstacle problems of the nonhomogeneous anisotropic elliptic equations under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe. [Copyright &y& Elsevier]

Abstract: The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians in with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains , , . Second, we extend Friedlander''s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems. [Copyright &y& Elsevier]

This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with L 2 boundary data in a periodically perforated Lipschitz domain. [ABSTRACT FROM AUTHOR]

In this paper, we study the vanishing dissipation limit of solutions to the initial boundary value problem of three-dimensional incompressible viscous magnetohydrodynamic equations in the upper half space where the magnetic field is assumed to be transverse to the boundary. When the tangential component of the magnetic field is imposed the zero Neumann boundary condition, we can establish the uniform energy estimates of the solutions to the initial boundary value problem in the conormal Sobolev spaces even if the velocity field satisfies the no-slip boundary condition. Then the vanishing dissipation limit of solutions to the initial boundary value problem can be achieved in L ∞ sense based on some compactness arguments. Compared with the previous results, our results show that the strong boundary layer can still be prevented by the transverse magnetic field even with the magnetic diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the free boundary value problem of the three-dimensional full primitive equations for the large-scale ocean with heat insulation in a bounded domain. The domain is periodic in the horizontal direction and finite in the vertical direction, and its upper boundary is a free surface driven by the fluid and bottom boundary is fixed. We show that the unique strong solution of the free boundary value problem for the three-dimensional full primitive equations exists globally in time and converges to the equilibrium state with exponential decay rate, when the initial data is a small perturbation of the steady state in regular Sobolev space. [ABSTRACT FROM AUTHOR]

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a diffusive system modeling the interaction of mussel and algae in the water layer overlying the mussel bed, where the algae are the main food source for mussels, and the advection pushes algaes in one direction but not out of the domain. We present a threshold result on the global extinction and persistence of mussels. It shows that the condition for persistence depends on the principal eigenvalue of a scalar boundary value problem, which is related to the diffusion, the speed of the tidal flow, the conversion rate of algae to mussel production, and mussel mortality rate. We further investigate the asymptotic profile of the positive steady state when it exists. [ABSTRACT FROM AUTHOR]

Morse index theory in the classical framework provides an equality between the spectral properties of the second variation of the Lagrangian functional and the oscillation properties of the space of solutions of the associated boundary value problem. For carrying over a similar result in the non-Hamiltonian context which can be useful for investigating the dynamical properties of dissipative systems, we introduce a new topological invariant, the so-called "degree-index" defined in terms of the Brouwer degree of a suitable determinant map of a boundary matrix, which provides one possible substitute of the Maslov index in this non-Hamiltonian framework and finally we prove the equality between the Morse index and the degree-index in this non-selfadjoint setting through a new abstract trace formula. In the last part of the paper we apply our theoretical results to some 1D reaction-diffusion systems. [ABSTRACT FROM AUTHOR]

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L 4 , 5), families of periodic orbits passing through collision, and orbits connecting L 4 to ejection or collision. [ABSTRACT FROM AUTHOR]

Abstract: We compare the above-mentioned article with the content of a previous publication. [Copyright &y& Elsevier]

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method is based on the Riemann boundary value problem on a bundle of three straight lines. Description of scattering problem data is given. [ABSTRACT FROM AUTHOR]

The paper is concerned with the stability property under perturbation Q → Q ˜ of different spectral characteristics of a boundary value problem associated in L 2 ([ 0 , 1 ] ; C 2) with the following 2 × 2 Dirac type equation (0.1) L U (Q) y = − i B − 1 y ′ + Q (x) y = λ y , B = ( b 1 0 0 b 2 ) , b 1 < 0 < b 2 , y = col (y 1 , y 2) , with a potential matrix Q ∈ L p ([ 0 , 1 ] ; C 2 × 2) and subject to the regular boundary conditions U y : = { U 1 , U 2 } y = 0. If b 2 = − b 1 = 1 this equation is equivalent to one dimensional Dirac equation. Our approach to the spectral stability relies on the existence of the triangular transformation operators for system (0.1) with Q ∈ L 1 , which was established in our previous works. The starting point of our investigation is the Lipshitz property of the mapping Q → K Q ± , where K Q ± are the kernels of transformation operators for system (0.1). Namely, we prove the following uniform estimate: ‖ K Q ± − K Q ˜ ± ‖ X ∞ , p 2 + ‖ K Q ± − K Q ˜ ± ‖ X 1 , p 2 ⩽ C ⋅ ‖ Q − Q ˜ ‖ p , Q , Q ˜ ∈ U p , r 2 × 2 , p ∈ [ 1 , ∞ ] , on balls U p , r 2 × 2 in L p ([ 0 , 1 ] ; C 2 × 2). It is new even for Q ˜ = 0. Here X ∞ , p 2 , X 1 , p 2 are the special Banach spaces naturally arising in such problems. We also obtained similar estimates for Fourier transforms of K Q ±. Both of these estimates are of independent interest and play a crucial role in the proofs of all spectral stability results discussed in the paper. For instance, as an immediate consequence of these estimates we get the Lipshitz property of the mapping Q → Φ Q (⋅ , λ) , where Φ Q (x , λ) is the fundamental matrix of the system (0.1). Assuming the spectrum Λ Q = { λ Q , n } n ∈ Z of L U (Q) to be asymptotically simple, denote by F Q = { f Q , n } | n | > N a sequence of corresponding normalized eigenvectors, L U (Q) f Q , n = λ Q , n f Q , n. Assuming boundary conditions (BC) to be strictly regular , we show that the mapping Q → Λ Q − Λ 0 sends L p ([ 0 , 1 ] ; C 2 × 2) either into ℓ p ′ or into the weighted ℓ p -space ℓ p ({ (1 + | n |) p − 2 }) ; we also establish its Lipshitz property on compact sets in L p ([ 0 , 1 ] ; C 2 × 2) , p ∈ [ 1 , 2 ]. The proof of the second estimate involves as an important ingredient inequality that generalizes classical Hardy-Littlewood inequality for Fourier coefficients. It is also shown that the mapping Q → F Q − F 0 sends L p ([ 0 , 1 ] ; C 2 × 2) into the space ℓ p ′ (Z ; C ([ 0 , 1 ] ; C 2) of sequences of continuous vector-functions, and has the Lipshitz property on compacts sets in L p ([ 0 , 1 ] ; C 2 × 2) , p ∈ [ 1 , 2 ]. Certain modifications of these spectral stability results are also proved for balls U p , r 2 × 2 in L p ([ 0 , 1 ] ; C 2 × 2) , p ∈ [ 1 , 2 ]. Note also that the proof of the Lipshitz property of the mapping Q → F Q − F 0 involves the deep Carleson-Hunt theorem for maximal Fourier transform, while the proof of this property for the mapping Q → Λ Q − Λ 0 relies on the estimates of the classical Fourier transform and is elementary in character. We apply our previous results to the damped string equation to establish the Riesz basis property and the asymptotic behavior of the eigenvalues of the corresponding dynamic generator under the assumptions d ∈ L 1 [ 0 , ℓ ] , ρ ∈ W 1 , 1 [ 0 , ℓ ] on the damping coefficient and the density of the string, that are weaker than previously treated in the literature. We also establish Lipshitz dependence on d and ρ in ℓ p -spaces of the remainders in the asymptotic formula for the eigenvalues. [ABSTRACT FROM AUTHOR]

In the paper we established the L 2 estimates for commutators of the Dirichlet-to-Neumann map generated by a divergence form parabolic operator, defined on R + n + 2 , with real, symmetric and t , λ − independent coefficient matrix, or more generally, a small complex L ∞ perturbation of such. The major new challenge, compared to our previous work in elliptic setting, is to handle the first order derivatives with respect to the time variable t which not only fall on solutions to parabolic equations but also on auxiliary functions. [ABSTRACT FROM AUTHOR]

Dynamics of the macroscopic motion of a compressible, viscous and heat-conducting, one-dimensional monatomic ionized gas can be described by a compressible Navier-Stokes type system. Compared with the corresponding compressible Navier-Stokes type systems modeling a viscous and heat conducting ideal polytropic gas [7,10,13–16,18,19,23,24,26,28,31,32] , the real gases [17] , nonlinear thermoviscoelasticity [5,6] , and a viscous radiative and reactive gas [11,12,20,21,30] , the dependence of the state of an ionized gas on the degree of ionization leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas makes it impossible to deduce the dissipative estimates on the first-order spatial derivative of the bulk velocity and the absolute temperature from the basic energy estimates obtained through the normalized entropy and, as a consequence, makes the construction of its global large solutions and the study of their large time behavior more difficult. By making full use of the intrinsic structure of equations, the existence of global large smooth solutions to two types of initial boundary value problems of one-dimensional, compressible Navier-Stokes system for a viscous and heat-conducting ionized gas is established in [22] for a class of density and/or temperature dependent transport coefficients, but since the estimates obtained there depend on the time variable, the problem on their large time behavior remains unsolved and the main purpose of this paper is to give the precise description of the large time behavior of the global solutions constructed in [22]. The key ingredient in our analysis is to deduce the uniform-in-time positive lower and upper bounds on the specific volume and the absolute temperature. [ABSTRACT FROM AUTHOR]

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art e N prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis. • General very high-order finite difference method which can be tuned to high-order accuracy with sparsity. • Adaptable numerical method to both global and local eigenvalue problem for discrete modes in linear stability theory. • Unified approach capable of computing discrete and continuous normal modes essential for bi-orthogonal decomposition. • Direct discretization on the non-uniform grid to achieve high-order accuracy while maintaining stencil stability. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions to an initial boundary value problem of the Hunter-Saxton equation in the half space when the viscosity tends to zero. By means of the asymptotic analysis with multiple scales, we first formally derive the equations for boundary layer profiles. Next, we study the well-posedness of the equations for the boundary layer profiles by using the compactness argument. Moreover, we construct an accurate approximate solution and use the energy method to obtain the convergence results of the vanishing viscosity limit. [ABSTRACT FROM AUTHOR]

This paper concerns the Ginzburg-Landau equation and the Meissner equations on a bounded multiply-connected domain in R 3. The novelty of these equations is that both of them contain an unknown potential and an unknown Neumann field as the Lagrange multipliers. The minimal and non-minimal solutions for each equation are obtained by variational methods, and the effect of the applied magnetic field on the Meissner states is examined. The Meissner A-system with potential is also examined and a solution is obtained by using the Mountain Pass Lemma of Ambrosetti and Rabinowitz. In our analysis the estimates of the natural boundary value problem of the Maxwell-Stokes system play an important role. [ABSTRACT FROM AUTHOR]

Developing surrogate models for costly mathematical models representing physical systems is challenging since it is typically not possible to generate large training data sets, i.e. to create a large experimental design. In such cases, it can be beneficial to constrain the surrogate approximation to adhere to the known physics of the model. This paper presents a novel methodology for the construction of physics-informed polynomial chaos expansions (PCE) that combines the conventional experimental design with additional constraints from the physics of the model represented by a set of differential equations and specified boundary conditions. A computationally efficient means of constructing physically constrained PCEs, termed PC2, are proposed and compared to the standard sparse PCE. Algorithms are presented for both full-order and sparse PC2 expansions and an iterative approach is proposed for addressing nonlinear differential equations. It is shown that the proposed algorithms lead to superior approximation accuracy and do not add significant computational burden over conventional PCE. Although the main purpose of the proposed method lies in combining training data and physical constraints, we show that the PC2 can also be constructed from differential equations and boundary conditions alone without requiring model evaluations. We further show that the constrained PCEs can be easily applied for uncertainty quantification through analytical post-processing of a reduced PCE by conditioning on the deterministic space-time variables. Several deterministic examples of increasing complexity are provided and the proposed method is demonstrated for uncertainty quantification. • PC2 – a novel framework for physically constrained polynomial chaos expansions is proposed. • An efficient algorithm based on constrained least squares and sparse solver is developed. • Analytical uncertainty quantification of approximated PDEs by PC2 is performed. [ABSTRACT FROM AUTHOR]

The Immersed Boundary (IB) method of Peskin (1977) [1] is useful for problems that involve fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each step in time, this method can be prohibitively inefficient without preconditioning. This work introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. In order to lay the groundwork for similar formulations of Stokes and Navier-Stokes equations, this paper focuses on the Poisson and Helmholtz equations to introduce the methodology and to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method without preconditioning. Furthermore, the iteration count is independent of both the mesh size and spacing of the immersed boundary points. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann boundary conditions. • Reformulated Immersed Boundary method for double layer integral equations. • Efficient IB constraint method for PDEs with prescribed boundary values. • Well-conditioned IB method resulting from second-kind integral equation formulation. • IB method for Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted. • Innovative Transparent Boundary Conditions: A novel transparent boundary condition is designed to enhance the accuracy and efficiency of simulations. • Enhanced Numerical Quadratures: Highly efficient and accurate numerical quadrature techniques are developed to ensure reliable computations. • Comprehensive Numerical Examples: An array of carefully chosen numerical examples are presented to demonstrate the method's effectiveness and versatility. [ABSTRACT FROM AUTHOR]

This paper is concerned with the local well-posedness for the free boundary value problem of smooth solutions to the cylindrical symmetric Euler equations with damping and related models, including the compressible Euler equations and the Euler-Poisson equations. The free boundary is moving in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. However, the compressible Euler equations or Euler-Poisson equations with damping become a degenerate system at the moving boundary. By setting a suitable weighted Sobolev space and using Hardy's inequality, we successfully overcome the singularity at the center point and the vacuum occurring on the moving boundary, and obtain the well-posedness of local smooth solutions. We also summarize the recent related results on the free boundary value problem for the Euler equations with damping, compressible Euler equations and Euler-Poisson equations. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of the energy associated to an initial boundary value problem involving a semilinear wave equation. The domain is an unbounded subset of R 2 with unbounded boundary. On the function responsible by the semilinearity we consider a very general decay rate control. The main tools are the Trudinger–Moser inequality combined with microlocal analysis arguments and the techniques of Lasiecka and Tataru. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems. [ABSTRACT FROM AUTHOR]

The initial boundary value problems for compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma are considered on exterior domains in this paper. With the radial symmetry assumption, the global existence of solutions to compressible Navier-Stokes-Poisson equations with the large initial data on a domain exterior to a ball in R n (n ≥ 1) is proved. Moreover, without any symmetry assumption, the global existence of smooth solutions near a given constant steady state for both compressible Navier-Stokes-Poisson and Euler-Poisson equations on an exterior domain in R 3 with physical boundary conditions is also established with the exponential stability. A key issue addressed in this paper is on the global-in-time regularity of solutions near physical boundaries. This is in particular so for the 3-D compressible Navier-Stokes-Poisson equations to which global smooth solutions of initial boundary value problems are seldom found in literature to the best of knowledge. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of solitary wave solutions of time-dependent nonlinear Schrödinger equations with behaviors tending to zero at infinity under certain conditions on trapping potentials and parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established. Particularly, for constant trapping potentials, we conclude that there is at most one radial ground state under certain conditions on parameters. [ABSTRACT FROM AUTHOR]

This paper discusses initial boundary value problem for a semilinear edge degenerate parabolic equation and corresponding stationary problem. We first find some initial conditions with different energy levels such that the solution exists globally and blows up in finite time, respectively. We also study the asymptotic behaviors like exponential decay and exponential growth for solution and energy function. Especially, we show the solution of evolution problem will converge to the steady state solution. Additionally, we find that there are two explicit vacuum regions which are ball and annulus respectively, that is to say, there is no solution belongs to them and all solutions are isolated by them. Finally, we discuss the existence of ground state solution to the stationary problem. The instability of the ground state solution is considered and we prove that there exists initial value such that the instability occurs as a blow-up in finite time. [ABSTRACT FROM AUTHOR]

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity. [ABSTRACT FROM AUTHOR]

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation ∂ t u − Δ e − Δ u + e − Δ u − 1 = 0. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects [6]. Our investigations reveal that we must control the size of both ‖ e − Δ u (x , 0) ‖ W 2 , 2 (Ω) and ‖ e Δ u (x , 0) ‖ ∞ , Ω suitably to achieve our results. Related results in [8,10] were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity. [ABSTRACT FROM AUTHOR]

Neural networks have emerged as a tool for solving differential equations in many branches of engineering and science. But their progress in frequency domain acoustics is limited by the vanishing gradient problem that occurs at higher frequencies. This paper discusses a formulation that can address this issue. The problem of solving the governing differential equation along with the boundary conditions is posed as an unconstrained optimization problem. The acoustic field is approximated to the output of a neural network which is constructed in such a way that it always satisfies the boundary conditions. The applicability of the formulation is demonstrated on popular problems in plane wave acoustic theory. The predicted solution from the neural network formulation is compared with those obtained from the analytical solution. A good agreement is observed between the two solutions. The method of transfer learning to calculate the particle velocity from the existing acoustic pressure field is demonstrated with and without mean flow effects. The sensitivity of the training process to the choice of the activation function and the number of collocation points is studied. • PINNs-based solution for problems in frequency domain acoustics is presented. • Problems associated with the existing Lagrange multiplier method are discussed. • Acoustic field is predicted in the ducts of two different cross-sections. • Mean flow and visco-thermal effects are captured with a maximum error of O (1 0 − 4). • Results show the ability of PINNs to emerge as an acoustic solver in the future. [ABSTRACT FROM AUTHOR]

Various techniques are developed for addressing the existence, uniqueness and numerical calculation of Wiener path integral (WPI) most probable path solutions. Specifically, the WPI technique for determining the stochastic response of diverse nonlinear dynamical systems treats the system response joint transition probability density function as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. This functional integral is evaluated, ordinarily, by resorting to an approximate approach that considers the contribution only of the most probable path. The most probable path corresponds to an extremum of the functional integrand and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). In this paper, first, it is shown that for the commonly considered case of the system nonlinearity being of polynomial form, there exist globally optimal solutions corresponding to the most probable path BVP. Further, relying on algebraic geometry concepts and tools, a condition is derived for determining if the BVP for the most probable path exhibits a unique solution over a specific region. Furthermore, a novel solution approach is developed for the BVP by relying on Sylvester's dialytic method of elimination. Notably, the method reduces the complexity of the BVP system of coupled multivariate polynomial equations by eliminating one or more variables. Various numerical examples pertaining to diverse nonlinear oscillators are included for demonstrating the capabilities of the developed techniques. [ABSTRACT FROM AUTHOR]

Let Ω be a C 2 bounded domain in R N (N ≥ 3), δ (x) = dist (x , ∂ Ω) and C H (Ω) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , (P ρ) where 0 < μ < C H (Ω) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 < p < N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem (P ρ) admits a positive solution if 0 < ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎. In this paper, we go further in the study of the solution set of (P ρ). We show that the problem admits at least two positive solutions if 0 < ρ < ρ ⁎ and a unique positive solution if ρ = ρ ⁎. We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic operators whose coefficients are signed measures. Our approach is probabilistic, which also gives a representation of the solution. The heat kernel estimates and the theory of additive functionals play a crucial role in our approach. [ABSTRACT FROM AUTHOR]

In this paper we prove an existence result for "solution obtained as limit of approximations" to a class of Dirichlet boundary value problems whose prototype is { − Δ p u = β (1 + | ∇ u |) q + c (x) | u | p − 2 u + f in Ω u = 0 on ∂ Ω , where Ω is a bounded open subset of R N , N ≥ 2 , 1 < p < 2 , Δ p u = div (| ∇ u | p − 2 ∇ u) , p − 1 < q ≤ p − 1 + p N , β is a positive constant, c with c ≥ 0 , c ≠ 0 and f are measurable functions satisfying suitable summability conditions depending on q. We further assume smallness assumptions on β , c and f. Our approach is based on Schauder's fixed point theorem. Similar results can be proved also for 2 ≤ p < N. [ABSTRACT FROM AUTHOR]

The notion of p-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of p -ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of p -ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity , rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral conditions. A similar picture emerges when p -ellipticity is considered. In this paper, we define three new notions of p -ellipticity, establish relationships between them and show that each of them does play an important role in solving boundary value problems. These important roles are demonstrated by establishing extrapolation results for solvability of the L p Dirichlet problem for elliptic systems, followed by applications of this result in two different scenarios: one for the Lamé system of linear elasticity and another in the theory of homogenization. [ABSTRACT FROM AUTHOR]

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value. [ABSTRACT FROM AUTHOR]

The paper investigates the existence, asymptotic boundary behavior and uniqueness of convex solutions to the Monge-Ampère equation that take infinite values on the boundary of smooth and bounded open convex sets in R n. [ABSTRACT FROM AUTHOR]

In this paper we consider the semilinear damped wave problem of the form { (α (t) u t) t − β (t) Δ u + γ (t) u t + δ (t) u = β (t) f (u) , x ∈ Ω , t > τ , u (x , t) = 0 , x ∈ ∂ Ω , t ⩾ τ , u (x , τ) = u τ (x) , u t (x , τ) = v τ (x) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space. [ABSTRACT FROM AUTHOR]

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in R 3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role. The solutions themselves are evaluated by means of Fourier expansions with respect to spherical harmonics. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a Banach space X into itself. The connection between the semigroup property and the major premise of Huygens' principle is emphasized. Another tool is a Drazin-like inverse operator A ad for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009). Unlike the latter two problems, the solution of Robin's problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet's and Neumann's problem do not work. The authors give several hints how to overcome these difficulties. [ABSTRACT FROM AUTHOR]

This paper considers compound operators on a general vector space. Explicit representations for these operators are obtained in various contexts with emphasis on the compound differential equations corresponding to a given linear differential equation. Applications to estimation of eigenvalues of self-adjoint boundary value problems and of the codimension of the stable manifold of a linear differential equation in a Banach space are given. [ABSTRACT FROM AUTHOR]

Motivated by various physical, cellular and ecological applications, there has been a recent resurgence of interest in studying the boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface, by using "bulk-surface models" (involving volumetric densities and surface densities) or by using models involving dynamical boundary conditions for volumetric densities. The surface in the models has no thickness. However, in some real world applications, the "surface" is actually a thin membrane, having positive, albeit small thickness δ , as in the cases of thermal barrier coatings for turbine engine blades and cell membranes. In such situations, rigorous derivations for these models seem lacking and desirable. In this paper, we start with two full models each of which contains reaction-diffusion equations in the bulk and the thin membrane, respectively, with two types of reasonable transmission conditions linking the two. Then in the limit of δ → 0 , we obtain two effective models, with one being a bulk-surface model and the other being the dynamical boundary value problem model, from which we can also recover the surface density of the other substance. Our analysis reveals that to have such effective models, it is crucial that (i) appropriate transmission conditions are satisfied on the interface between the bulk and the thin membrane, and (ii) the diffusion tensor in the thin membrane is "optimally aligned", which means the principal axes of the tensor are either normal or tangential to the interface. [ABSTRACT FROM AUTHOR]

This paper is concerned with an initial and boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients. In the case when the viscosity μ (θ) = θ α and the heat-conductivity κ (θ) = θ β with α , β ∈ 0 , ∞) , we prove the global-in-time existence of strong solutions under some assumptions on the growth exponent α and the initial data. As a byproduct, the nonlinearly exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if α ≥ 0 is small, and the growth exponent β ≥ 0 can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper we prove uniqueness in the inverse boundary value problem for quasilinear elliptic equations whose linear part is the Laplacian and nonlinear part is the divergence of a function analytic in the gradient of the solution. The main novelty in terms of the result is that the coefficients of the nonlinearity are allowed to be "anisotropic". As in previous works, the proof reduces to an integral identity involving the tensor product of the gradients of 3 or more harmonic functions. Employing a construction method using Gaussian quasi-modes, we obtain a convenient family of harmonic functions to plug into the integral identity and establish our result. [ABSTRACT FROM AUTHOR]