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This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [ABSTRACT FROM AUTHOR]

In this paper we study a free boundary problem modeling the growth of vascularized tumors. The model is a modification to the Byrne–Chaplain tumor model that has been intensively studied during the past two decades. The modification is made by replacing the Dirichlet boundary value condition with the Robin condition, which causes some new difficulties in making rigorous analysis of the model, particularly on existence and uniqueness of a radial stationary solution. In this paper we successfully solve this problem. We prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient, whereas unstable for small surface tension coefficient. Tools used in this analysis are the geometric theory of abstract parabolic differential equations in Banach spaces and spectral analysis of the linearized operator. [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]

In this paper we continue to deal with the initial–boundary value problem for the coupled Keller–Segel–Stokes system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , ( x , t ) ∈ Ω × ( 0 , T ) , c t + u ⋅ ∇ c = Δ c − c + n , ( x , t ) ∈ Ω × ( 0 , T ) , u t + ∇ P = Δ u + n ∇ ϕ , ( x , t ) ∈ Ω × ( 0 , T ) , ∇ ⋅ u = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , where Ω ⊂ R d is a bounded domain with smooth boundary and the chemotactic sensitivity S is not a scalar function but rather attains values in R d × d , and satisfies | S ( x , n , c ) | ≤ C S ( 1 + n ) − α with some C S > 0 and α > 0 . When d = 2 , our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption α > 0 , which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting. In this paper, for the case d = 3 , we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption α > 1 2 , which is slightly stronger than the corresponding subcritical assumption α > 1 3 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general L p regularity of u from a rather low L p regularity of n , which will be obtained by a new combinational functional. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the quasilinear fully parabolic Keller–Segel system under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary, . The diffusivity is assumed to satisfy some further technical conditions such as algebraic growth and , which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that for with , and . When , this paper represents an improvement of Tao and Winkler [17], because the domain does not necessarily need to be convex in this paper. In the case and , uniform-in-time boundedness is an open problem left in a previous paper [7]. This paper also gives an answer to it in bounded domains. [Copyright &y& Elsevier]

Abstract: In this paper we study a free boundary problem modeling solid avascular tumor growth. The model is based on the reaction–diffusion dynamics and mass conservation law. The model is considered with time delays in proliferation process. The quasi-steady-state (i.e., ) is studied by Foryś and Bodnar [see U. Foryś, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling 37 (2003) 1201–1209]. In this paper we mainly consider the case . In the case considered by Foryś and Bodnar, the model is reduced to an ordinary differential equation with time delay, but in the case the model cannot be reduced to an ordinary differential equation with time delay. By theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the long time asymptotic behavior of the solutions under some conditions. [Copyright &y& Elsevier]

Abstract An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed_dashpot, fixed_spring-dashpot, fixed_mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark- β method solution. The energy expressions for a travelling string with a fixed_dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates. [ABSTRACT FROM AUTHOR]

Abstract The motion of moderately thick spray can be modeled by a coupled system of equations consisting of the incompressible Navier–Stokes equations and the Vlasov–Boltzmann equation. In this paper, we study the initial value problem for the Navier–Stokes–Vlasov–Boltzmann equations. The existence of global weak solutions is established by the weak convergence method. [ABSTRACT FROM AUTHOR]

We consider a thin-film equation. By exploiting the boundary condition and the variational structure of the equation, we consider the global existence and blow-up of the solutions. For the global solutions, we study the asymptotic behavior, and for the blow-up solutions, we study the upper and lower bounds of the blow-up time. Especially, we have derived the necessary and sufficient conditions for the solution blowing up in finite time when the initial energy J ( u 0 ) ≤ d , where d is the mountain-pass level. For J ( u 0 ) > d , we also study the conditions for the solution exists globally and blows up in finite time, and we also obtain some necessary and sufficient conditions for the solution blowing up in finite time when the initial energy at arbitrary level. Furthermore, we study the decay behavior of both the global solutions and its corresponding energy functional, and the decay ratios are given specially. The results generalize the former studies on this equation, such as the papers: Li et al. (2016) [16] , Dong and J. Zhou (2017) [4] and Sun et al. (2018) [23] . [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the study of nonlinear singular and non-singular fourth order difference equations coupled with periodic boundary value conditions. In the paper some existence and nonexistence results are given. The results are deduced from Krasnoselskii''s fixed point theorems in cones. An exhaustive study of the Green''s function related to a linear fourth order problem is done. [Copyright &y& Elsevier]

Abstract: In the beginning of the 1990s we devoted a sequence of papers to perturbation theory, singular limits and well-posedness problems. In particular, the strong well-posedness of the initial–boundary value problem for the compressible Euler equations was demonstrate for the first time. Our method also allowed singular limit results in the strong norm, even under assumptions weaker than the current ones in the literature (where the strong norm is not reached). It is worth noting that, until now, the above method and results have not been substantially improved. Hence an introduction to it still looks timely. Actually, in a forthcoming paper, by returning to this method, we improve (in a very substantial way) some important results recently appeared in the literature. [Copyright &y& Elsevier]

Abstract: We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28 (2005) 917–945] where an energy a priori estimate for the linearized free boundary value problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in [S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622–629 (in Russian); W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655]. In this paper we derive an a priori estimate in the whole stability domain. The crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss'' type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition, i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of derivatives in the energy estimate. The result of this paper is a necessary step to prove the local-in-time existence of stable nonplanar incompressible current-vortex sheets by a suitable Nash–Moser type iteration scheme. [Copyright &y& Elsevier]

This paper deals with the free transverse vibration of a non-homogeneous tapered beam subjected to general axial forces, with arbitrarily located internal hinge and elastics supports, and ends elastically restrained against rotation and translation. A rigorous and complete development is presented. First, a brief description of several papers previously published is included. Second, the Hamilton principle is rigorously stated by defining the domain D of the action integral and the space Da of admissible directions. The differential equations, boundary conditions, and particularly the transitions conditions, are obtained. Third, the transition conditions are analysed for several sets of restraints conditions. Fourth, the existence and uniqueness of the weak solutions of the boundary value problem and the eigenvalue problem which, respectively, govern the statical and dynamical behaviour of the mentioned beam is treated. Finally, the method of separation of variables is used for the determination of the exact frequencies and mode shapes and a modern application of the Ritz method to obtain approximate eigenvalues. In order to obtain an indication of the accuracy of the developed mathematical model, some cases available in the literature have been considered. New results are presented for different boundary conditions and restraint conditions in the internal hinge. [Copyright &y& Elsevier]

Abstract: In this paper, the initial value problem for a class of fractional differential equations is discussed, which generalizes the existent result to a wide class of fractional differential equations. Also the theoretical result established in the paper ensures the validity of chaos control of fractional differential equations. In particular, feed-back control of chaotic fractional differential equation is theoretically investigated and the fractional Lorenz system as a numerical example is further provided to verify the analytical result. [Copyright &y& Elsevier]

Abstract: Based on the Flügge thin shell theory, this paper presents exact solutions for the vibration of circular cylindrical shells with step-wise thickness variations in the axial direction. The shell is sub-divided into multiple segments at the locations of thickness variations. The state-space technique is adopted to derive the homogenous differential equations for a shell segment and domain decomposition method is employed to impose the equilibrium and compatibility requirements along the interfaces of the shell segments. To ensure the correctness of the present results, comparisons are made with one paper available in the open literature based on the Donnell–Mushtari theory. Shells with various combinations of end boundary conditions can be analyzed by the proposed method. Furthermore, the influences of the shell thickness ratios, locations of step-wise thickness variations and step thickness ratios on the natural frequencies and mode shapes are examined. The exact vibration results can serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shells. [Copyright &y& Elsevier]

Abstract: In this paper we prove that the spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof. [Copyright &y& Elsevier]

Abstract: In this paper, we develop an efficient preconditioning method on the basis of the modified hierarchy basis for solving the singular boundary value problem by the Galerkin method. After applying the preconditioning method, we show that the condition number of the linear system arising from the Galerkin method is uniformly bounded. In particular, the condition number of the preconditioned system will be bounded by 2 for the case (see Eq. (1) in the paper). Numerical results are presented to confirm our theoretical results. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of boundary value problem of singular differential equations on time scales. The conditions we used here differ from those in the majority of papers as we know. An existence theorem of positive solutions is established by using the Krasnosel''skii fixed point theorem and an example is given to illustrate it. [Copyright &y& Elsevier]

Abstract: Let R be a local ring essentially of finite type over a field k of characteristic . In the paper [M. Furuya, H. Niitsuma, Regularity criterion of Noetherian local rings of prime characteristic, J. Algebra 247 (2002) 219–230], we constructed some regularity criterion for such a local ring R in terms of the higher differential algebra and the -basis. In this paper, we introduce the concept of a reduced index of a Noetherian ring and we show the sharpened result of the above criterion and further we also give a geometric regularity criterion in terms of the higher differential algebra and the -basis. The latter criterion yield the sharpened result of a part of Orbanz''s theorem [U. Orbanz, Höhere Derivationen und Regularität, J. Reine. Angew. Math. 262/263 (1973) 194–204, 4.2]. [Copyright &y& Elsevier]

Abstract: We consider a 2mth order elliptic operator of divergence form in a domain Ω of , whose leading coefficients are uniformly continuous. In the paper [Y. Miyazaki, The theory of divergence form elliptic operators under the Dirichlet condition, J. Differential Equations 215 (2005) 320–356], we developed the theory including the construction of resolvents, assuming that the boundary of Ω is of class . The purpose of this paper is to show that the theory also holds when Ω is a domain, applying the inequalities of Hardy type for the Sobolev spaces. [Copyright &y& Elsevier]

Abstract: This is the second paper of a series in which we build and study third-order Hermitian upwind schemes for numerically solving linear aeroacoustics. In this paper, we extend the method to non-stationary flows. For this purpose, we employ a family of six-wave models to discretize the space operator. This family is parametrized by an “acoustic propagation angle”. From this wave modelling, specific boundary conditions are proposed to treat effectively subsonic/supersonic boundary conditions. A sequence of numerical simulations is then carried out and makes it possible to examine the effectiveness of the scheme. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationwhere the nonlinear term depends on the unknown function u and its derivative and satisfy the estimatewith such that The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions. [Copyright &y& Elsevier]

Abstract: The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state. To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold. [Copyright &y& Elsevier]

Abstract: In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states at a nonlinear eigenvalue with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz''s unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz''s context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181). [Copyright &y& Elsevier]

Abstract: Deriving high precision schemes to compute turbulent flows on fixed or moving complex grids is becoming a central issue in the direct numerical simulation (DNS) and large eddy simulation (LES) community. The step between classical DNS/LES codes on fixed structured grids and future methods on moving unstructured grids is a significant evolution in terms of numerical methods. For reacting flows, this evolution must also include more precise descriptions of multispecies flows and boundary conditions. This paper describes the development of a method for unsteady multispecies reacting flows on moving grids. The target field of application of this method is DNS and LES but this paper focuses on the method development and elementary test cases. The theoretical basis for the numerical method, the boundary conditions and the moving grid extension are first discussed. Various tests of the method are then provided on fixed and moving grids for simple reacting and non-reacting flows to demonstrate the precision and power of the method in simple reference laminar cases. [Copyright &y& Elsevier]

This paper is concerned with the asymptotic behavior of solution to the Cahn–Hilliard equation subject to the following dynamic boundary conditions: and the initial condition where Ω is a bounded domain in Rn (n⩽3) with smooth boundary Γ , and Γs>0, σs>0, gs>0, hs are given constants; Δ|| is the tangential Laplacian operator, and ν is the outward normal direction to the boundary.This problem has been considered in the recent paper by Racke and Zheng (Adv. Differential Equations 8 (1) (2003) 83) where the global existence and uniqueness were proved. In a very recent manuscript by Prüss, Racke and Zheng (Konstanzer Schrift. Math. Inform. 189 (2003)) the results on existence of global attractor and maximal regularity of solution have been obtained. In this paper, convergence of solution of this problem to an equilibrium, as time goes to infinity, is proved. [Copyright &y& Elsevier]

In this paper, the aim is to find the non-linear mode shapes and natural frequencies for a class of one-dimensional continuous damped systems with weak cubic inertia, damping and stiffness non-linearities. This paper presents general formulations for natural frequencies and mode shapes with all non-linearity effects. Initially the non-linear system with general boundary conditions is discretized, and using a two-dimensional manifold, the model of cubic non-linearities is constructed and the general equation of motion which governs non-linear system is derived. The method of multiple scales is then used to extend the non-linear mode shapes and natural frequencies. During this analysis, it is realized that when the natural frequencies of the linear system become equal to the natural frequencies of the non-linear system a one-to-one internal resonance will appear. Also, there is a three-to-one internal resonance which is not dependent on the damping of the system. Finally, general formulations of amplitude for vibrations, natural frequencies and mode shapes of the non-linear system are obtained in parametric forms. Thus, a non-linear problem with some simple integration can be solved. The formulations are capable of handling any non-linearities in inertia, damping, stiffness, or any combination of them under any arbitrary boundary conditions. [Copyright &y& Elsevier]

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper. [Copyright &y& Elsevier]

We consider in this paper the mathematical and numerical modeling of reflective boundary conditions (BC) associated to Boltzmann–Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability p ( k → ) . We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of boundary value problems for p -Laplacian equation ( Φ p ( u ′ ) ) ′ + h ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 with integral boundary conditions u ( 0 ) − α u ′ ( 0 ) = ∫ 0 1 g 1 ( s ) u ( s ) d s , u ( 1 ) + β u ′ ( 1 ) = ∫ 0 1 g 2 ( s ) u ( s ) d s . By using the Avery–Peterson fixed point theorem, we obtain the existence of at least three positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial-boundary value problem for a generalized Kelvin–Voight equation with p -Laplacian and a damping term: v → t + ( v → ⋅ ∇ ) v → + ∇ P ( x , t ) = ϰ div ( ∇ v → t ) + ν div ( | ∇ v → | p − 2 ∇ v → ) + γ | v → | m − 2 v → , div v → = 0 . Here v → ( x , t ) is the velocity field, P ( x , t ) is the pressure, ν is the viscosity kinematic coefficient, and ϰ is the viscosity relaxation coefficient (is a length scale parameter characterizing the elasticity of the fluid). The coefficient γ and the exponents p , m are given constants. Under appropriate conditions on the data, we prove the existence and uniqueness of the global and local weak solutions. Under several assumptions on the exponents p , m , the coefficients ν , ϰ , and specified initial data, a finite time blow-up and the behavior of the solutions for large times are also established. [ABSTRACT FROM AUTHOR]

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary . We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ { p 0 } ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached. [ABSTRACT FROM AUTHOR]

This paper analyzes the mechanical and mathematical models in “Ritto et al. (2013) [1]”. The results are that: (1) the mechanical model is obviously incorrect; (2) the mathematical model is not complete; (3) the differential equation is obviously incorrect; (4) the finite element equation is obviously not discretized from the corresponding mathematical model above, and is obviously incorrect. A mathematical model of dynamics should include the differential equations, the boundary conditions and the initial conditions. [ABSTRACT FROM AUTHOR]

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L 2 ( − 1 , 1 ) which has the Legendre polynomials { P n } n = 0 ∞ as eigenfunctions. As a consequence, they explicitly determined the domain D ( A 2 ) of the self-adjoint operator A 2 . However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point x = ± 1 in L 2 ( − 1 , 1 ) so D ( A 2 ) should exhibit four boundary conditions. In this paper, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman–Krein–Naimark) theory. In addition, we determine a new characterization of D ( A 2 ) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple – and natural – and are equivalent to the boundary conditions obtained from the GKN theory. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamics of parabolic semilinear differential equations with homogeneous Dirichlet boundary conditions via the discretization of finite element method. We provide an appropriate functional setting to treat this problem and, as a first step, we show the continuity of the set of equilibria and of its linear unstable manifolds. [ABSTRACT FROM AUTHOR]

In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Łojasiewicz–Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated L 2 -gradient flow converges for large time to an elastica, that is to a critical point of the functional. [ABSTRACT FROM AUTHOR]

Many wave propagation problems (in acoustics or in railway catenary or cable car dynamics, for example) can be solved with high efficiency if the computational domain can be truncated to a small region of interest with appropriate absorbing boundary conditions. In this paper, highly absorbing and stable boundary conditions for linear partial differential equations discretized by finite difference schemes are directly designed using a flexible, optimization-based formulation. The proposed optimization approach to the computation of the absorbing boundary conditions is capable of optimizing the accuracy (the absorbing quality of the boundary condition) while guaranteeing stability of the discretized partial differential equations with the absorbing boundary conditions in place. Penalty functions are proposed that explicitly quantify errors introduced by the boundary condition on the solution of the bounded domain compared to the solution of the unbounded domain problem. Together with the stability condition the described approach can be applied on various types of linear partial differential equations and is thus applicable for generic wave propagation problems. Its flexibility and efficiency is demonstrated for two engineering problems: The Euler–Bernoulli beam under axial load, which can be used to model cables as well as catenary flexural dynamics, and a two-dimensional wave as commonly encountered in acoustics. The accuracy of the absorbing boundary conditions obtained by the proposed concept is compared to analytical absorbing boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we study the following Hénon-like equation { − Δ u = | | y | − 2 | α u p , u > 0 , in Ω , u = 0 , on ∂ Ω , where α > 0 , p = N + 2 N − 2 , Ω = { y ∈ R N : 1 < | y | < 3 } , N ≥ 4 . We show that for α > 0 the above problem has infinitely many positive solutions concentrating simultaneously near the interior boundary { x ∈ R N : | x | = 1 } and the outward boundary { x ∈ R N : | x | = 3 } , whose energy can be made arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial–boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H 2 initial data. [ABSTRACT FROM AUTHOR]

Abstract: An exact expression of steady discrete shocks was recently obtained by the author in [9] for a class of residual-based compact schemes (RBC) applied to the inviscid Bürgers equation in a finite domain. Following the same lines, the analysis is extended to an infinite domain for a scalar conservation law with a general convex flux. For the dissipative high-order schemes considered, discrete shocks in infinite domain or with boundary conditions at short distance (Rankine–Hugoniot relations) are found to be very close. Besides, the present analytical description of shock capturing in infinite domain is explicit and so simple that it could lead to a new approach for correcting parasitic oscillations of high order RBC schemes. In a second part of the paper, exact solutions are also derived for equivalent differential equations (EDE) approximating schemes (subscript denotes the accuracy order) at orders 2p and . Although EDE involves Taylor expansions around steep structures, agreement between the exact EDE shock-profiles and the discrete shocks is remarkably good for and schemes. In addition, a strong similarity is demonstrated between the analytical expressions of discrete shocks and EDE shock profiles. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to extend to the setting of locally convex spaces the study of the mixed variational formulation of some elliptic boundary value problems, the so-called Babuška–Brezzi theory. This study consists of characterizing the existence of a solution and giving conditions that guarantee the stability of the corresponding Galerkin method. [Copyright &y& Elsevier]

Abstract: This paper introduces a formalism which extends that of “Green's function” and that of “the Volterra series”. These formalisms are typically used to solve, respectively, linear inhomogeneous space–time differential equations in physics and weakly nonlinear time-differential input-to-output systems in automatic control. While Green's function is a space–time integral kernel which fully characterizes a linear problem, the Volterra series expansions involve a sequence of multi-variate time integral kernels (of convolution type for time-invariant systems). The extension proposed here consists in combining the two approaches, by introducing a series expansion based on multi-variate space–time integral kernels. This series allows the representation of the space–time solution of weakly nonlinear boundary problems excited by an “input” which depends on space and time. This formalism is introduced on and applied to a nonlinear model of a damped string that is excited by a transverse mass force . The Green–Volterra kernels that solve the transverse displacement dynamics are computed. The first-order kernel exactly corresponds to Green's function of the linearized problem. The higher order kernels satisfy a sequence of linear boundary problems that lead to (both) analytic closed-form solutions and modal decompositions. These results lead to an efficient simulation structure, which proves to be as simple as the one based on the Volterra series, that has been obtained in a previous work for excitation forces with separated variables . Numerical results are presented. [Copyright &y& Elsevier]

Abstract: In this paper, an exact dynamic stiffness formulation using one-dimensional (1D) higher-order theories is presented and subsequently used to investigate the free vibration characteristics of solid and thin-walled structures. Higher-order kinematic fields are developed using the Carrera Unified Formulation, which allows for straightforward implementation of any-order theory without the need for ad hoc formulations. Classical beam theories (Euler–Bernoulli and Timoshenko) are also captured from the formulation as degenerate cases. The Principle of Virtual Displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to carry out the free vibration analysis of solid and thin-walled structures. The accuracy of the theory is confirmed both by published literature and by extensive finite element solutions using the commercial code . [Copyright &y& Elsevier]

Abstract: This paper is devoted to modelling vibration of layered curved circular arch with perfect or imperfect bonding between any two adjacent layers. The author has developed an efficient iterative process to obtain accurate determination of the stress and displacement fields for “stress critical” calculations such as damping and delamination. The differential equations which govern the free vibrations of a circular ring segment and the associated boundary conditions are derived by Hamilton's principle considering shear deformation and normal deformations of all layers. The interfacial perfectly or weakly bonding conditions and free traction conditions on the lateral surfaces are ensured. For the imperfect bonding, a general spring-layer model is adopted. The author used a new iterative process to successively refine the stress/strain field in the sandwich arch. The new models are used to predict the modal frequencies and damping of layered curved arch with perfect or imperfect bonding between any two adjacent layers. The solutions for a three layer circular arch are compared to a three layer approximate model. [Copyright &y& Elsevier]

Abstract: In this paper, non-self-adjoint Sturm–Liouville operators in Weyl’s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm–Liouville differential expression. Then, using the characteristic determinant, the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the periodic boundary value problem where is a given constant, is a parameter, and satisfies for all . The variational principle is given and some multiplicity results of periodic solutions of (1) are obtained via variational methods. [Copyright &y& Elsevier]

Abstract: In this paper, the -index is defined and the -boundary value problem of Hamiltonian system is studied. As applications, the existence and multiplicity results of periodic solutions of asymptotically linear delay differential systems and delay Hamiltonian systems are obtained. [Copyright &y& Elsevier]