Authors:Sisi Song Abstract: This paper concerns the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum on \(\varOmega \subset \mathbb {R}^3\) . The domain \(\varOmega \subset \mathbb {R}^3\) is a general connected smooth one, either bounded or unbounded. In particular, the initial density can have compact support when \(\varOmega \) is unbounded. First, we obtain the local existence and uniqueness of strong solution to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations without any compatibility condition assumed on the initial data. Then, we also prove the continuous dependence of strong solution on the initial data under an additional compatibility condition. PubDate: 2018-02-07 DOI: 10.1007/s00033-018-0915-z Issue No:Vol. 69, No. 2 (2018)

Authors:Xiaopeng Zhao; Mingxuan Zhu Abstract: In this paper, we consider the small initial data global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in \(\mathbb {R}^3\) . In addition, we also establish the temporal decay estimates for the weak solutions. With these estimates in hand, we study the algebraic time decay for higher-order Sobolev norms of small initial data solutions. PubDate: 2018-02-06 DOI: 10.1007/s00033-018-0907-z Issue No:Vol. 69, No. 2 (2018)

Authors:Shashi Kant; Manushi Gupta; Om Namha Shivay; Santwana Mukhopadhyay Abstract: In this work, we consider a two-dimensional dynamical problem of an infinite space with finite linear Mode-I crack and employ a recently proposed heat conduction model: an exact heat conduction with a single delay term. The thermoelastic medium is taken to be homogeneous and isotropic. However, the boundary of the crack is subjected to a prescribed temperature and stress distributions. The Fourier and Laplace transform techniques are used to solve the problem. Mathematical modeling of the present problem reduces the solution of the problem into the solution of a system of four dual integral equations. The solution of these equations is equivalent to the solution of the Fredholm’s integral equation of the first kind which has been solved by using the regularization method. Inverse Laplace transform is carried out by using the Bellman method, and we obtain the numerical solution for all the physical field variables in the physical domain. Results are shown graphically, and we highlight the effects of the presence of crack in the behavior of thermoelastic interactions inside the medium in the present context, and its results are compared with the results of the thermoelasticity of type-III. PubDate: 2018-02-02 DOI: 10.1007/s00033-018-0914-0 Issue No:Vol. 69, No. 2 (2018)

Authors:Miroslav Bulíček; Jan Burczak Abstract: We consider a model for deformations of a homogeneous isotropic body, whose shear modulus remains constant, but its bulk modulus can be a highly nonlinear function. We show that for a general class of such models, in an arbitrary space dimension, the respective PDE problem has a unique solution. Moreover, this solution enjoys interior smoothness. This is the first full regularity result for elasticity problems that covers the most natural space dimension 3 and that captures the behaviour of real-life elastic materials (considered in small deformations), primarily certain beta-phase titanium alloys. PubDate: 2018-01-31 DOI: 10.1007/s00033-018-0917-x Issue No:Vol. 69, No. 1 (2018)

Authors:S. R. Holcombe Abstract: A method for evaluating finite trigonometric summations is applied to a system of N coupled oscillators under acceleration. Initial motion of the nth particle is shown to be of the order \(T^{2{n}+2}\) for small time T, and the end particle in the continuum limit is shown to initially remain stationary for the time it takes a wavefront to reach it. The average velocities of particles at the ends of the system are shown to take discrete values in a step-like manner. PubDate: 2018-01-29 DOI: 10.1007/s00033-018-0911-3 Issue No:Vol. 69, No. 1 (2018)

Authors:Yazdan Hayati; Morteza Eskandari-Ghadi Abstract: An asymmetric three-dimensional thermoelastodynamic wave propagation with scalar potential functions is presented for an isotropic half-space, in such a way that the wave may be originated from an arbitrary either traction or heat flux applied on a patch at the free surface of the half-space. The displacements, stresses and temperature are presented within the framework of Biot’s coupled thermoelasticity formulations. By employing a complete representation for the displacement and temperature fields in terms of two scalar potential functions, the governing equations of coupled thermoelasticity are uncoupled into a sixth- and a second-order partial differential equation in cylindrical coordinate system. By virtue of Fourier expansion and Hankel integral transforms, the angular and radial variables are suppressed respectively, and a \(6{\mathrm{th}}\) - and a \(2{\mathrm{nd}}\) -order ordinary differential equation in terms of depth are received, which are solved readily, from which the displacement, stresses and temperature fields are derived in transformed space by satisfying both the regularity and boundary conditions. By applying the inverse Hankel integral transforms, the displacements and temperature are numerically evaluated to determine the solutions in the real space. The numerical evaluations are done for three specific cases of vertical and horizontal time-harmonic patch traction and a constant heat flux passing through a circular disc on the surface of the half-space. It has been previously proved that the potential functions used in this paper are applicable from elastostatics to thermoelastodynamics. Thus, the analytical solutions presented in this paper are verified by comparing the results of this study with two specific problems reported in the literature, which are an elastodynamic problem and an axisymmetric quasi-static thermoelastic problem. To show the accuracy of numerical results, the solution of this study is also compared with the solution for elastodynamics exists in the literature for surface excitation, where a very good agreement is achieved. The formulations presented in this study may be used as benchmark for other related researches and it may be implemented in the related boundary integral equations. PubDate: 2018-01-29 DOI: 10.1007/s00033-018-0910-4 Issue No:Vol. 69, No. 1 (2018)

Authors:Alla V. Ilyashenko; Sergey V. Kuznetsov Abstract: The dispersion relations are derived for SH waves in stratified anisotropic plates with arbitrary elastic anisotropy. Analytical expressions for vectorial group and ray velocities of SH waves propagating in anisotropic layers with monoclinic symmetry are obtained. Closed-form relations between velocities and specific kinetic and strain energy for SH waves are derived and analyzed. PubDate: 2018-01-25 DOI: 10.1007/s00033-018-0916-y Issue No:Vol. 69, No. 1 (2018)

Authors:Basant Lal Sharma Abstract: An exact expression for the scattering matrix associated with a junction generated by partial unzipping along the zigzag direction of armchair tubes is presented. The assumed simple, but representative, model, for scalar wave transmission can be interpreted in terms of the transport of the out-of-plane phonons in the ribbon-side vis-a-vis the radial phonons in the tubular-side of junction, based on the nearest-neighbor interactions between lattice sites. The exact solution for the ‘bondlength’ in ‘broken’ versus intact bonds can be constructed via a standard application of the Wiener–Hopf technique. The amplitude distribution of outgoing phonons, far away from the junction on either side of it, is obtained in closed form by the mode-matching method; eventually, this leads to the provision of the scattering matrix. As the main result of the paper, a succinct and closed form expression for the accompanying reflection and transmission coefficients is provided along with a detailed derivation using the Chebyshev polynomials. Applications of the analysis presented in this paper include linear wave transmission in nanotubes, nanoribbons, and monolayers of honeycomb lattices containing carbon-like units. PubDate: 2018-01-25 DOI: 10.1007/s00033-018-0909-x Issue No:Vol. 69, No. 1 (2018)

Authors:Qiang Tao; Ying Yang; Jincheng Gao Abstract: In this paper, we study the existence and uniqueness of the global classical solution for the planar compressible Hall-magnetohydrodynamic equations with large initial data. The system is supplemented with free boundary and smooth initial conditions. The proof relies on the bounds of the density and the skew-symmetric structure of the Hall term. PubDate: 2018-01-24 DOI: 10.1007/s00033-018-0912-2 Issue No:Vol. 69, No. 1 (2018)

Authors:Azer Khanmamedov; Sema Yayla Abstract: We consider the initial boundary value problem for the hyperbolic relaxation of the 2D Cahn–Hilliard equation with sub-cubic nonlinearity. Under mild regularity conditions on the nonlinearity, we prove the uniform (with respect to the initial data) boundedness of the weak solutions without assuming lower bound condition on the first derivative of the nonlinear term. Then, we prove the existence of the regular global attractor for the weak solutions. PubDate: 2018-01-22 DOI: 10.1007/s00033-018-0908-y Issue No:Vol. 69, No. 1 (2018)

Authors:Duc Chinh Pham Abstract: Variational results on the macroscopic conductivity (thermal, electrical, etc.) of the multi-coated sphere assemblage have been used to derive the explicit expression of the respective field (thermal, electrical, etc.) within the spheres in d dimensions ( \(d=2,3\) ). A differential substitution approach has been developed to construct various explicit expressions or determining equations for the effective spherically symmetric inclusion problems, which include those with radially variable conductivity, different radially variable transverse and normal conductivities, and those involving imperfect interfaces, in d dimensions. When the volume proportion of the outermost spherical shell increases toward 1, one obtains the respective exact results for the most important specific cases: the dilute solutions for the compound inhomogeneities suspended in a major matrix phase. Those dilute solution results are also needed for other effective medium approximation schemes. PubDate: 2018-01-11 DOI: 10.1007/s00033-017-0905-6 Issue No:Vol. 69, No. 1 (2018)

Authors:Zhen-Hui Bu; Zhi-Cheng Wang Abstract: This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP monostable equations. Our study contains two parts: in the first part, we first show that the two-dimensional V-shaped traveling fronts are asymptotically stable in \(\mathbb {R}^{n+2}\) with \(n\ge 1\) under any (possibly large) initial perturbations that decay at space infinity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation. In the second part, we establish the multidimensional stability of planar traveling front in \(\mathbb {R}^{n+1}\) with \(n\ge 1\) . PubDate: 2018-01-10 DOI: 10.1007/s00033-017-0906-5 Issue No:Vol. 69, No. 1 (2018)

Authors:Jian-Wen Sun Abstract: In this paper, we consider the positive solutions of the nonlocal dispersal equation $$\begin{aligned} \int \limits _{\Omega }J(x,y)[u(y)-u(x)]\mathrm{d}y=-\lambda m(x)u(x)+[c(x)+\varepsilon ]u^p(x) \quad \text { in }\bar{\Omega }, \end{aligned}$$ where \(\Omega \subset \mathbb {R}^N\) is a bounded domain, \(\lambda ,\varepsilon \) and \(p>1\) are positive constants. The dispersal kernel J and the coefficient c(x) are nonnegative, but c(x) has a degeneracy in some subdomain of \(\Omega \) . In order to study the influence of heterogeneous environment on the nonlocal system, we study the sharp spatial patterns of positive solutions as \(\varepsilon \rightarrow 0\) . We obtain that the positive solutions always have blow-up asymptotic profiles in \(\bar{\Omega }\) . Meanwhile, we find that the profiles in degeneracy domain are different from the domain without degeneracy. PubDate: 2017-12-18 DOI: 10.1007/s00033-017-0903-8 Issue No:Vol. 69, No. 1 (2017)

Authors:Shangjiang Guo Abstract: In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous) steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet/Neumann boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to population models with one-dimensional spatial domain. PubDate: 2017-12-15 DOI: 10.1007/s00033-017-0904-7 Issue No:Vol. 69, No. 1 (2017)

Authors:Asha Kumari Meena; Harish Kumar Abstract: In this article, we consider the Ten-Moment equations with source term, which occurs in many applications related to plasma flows. We present a well-balanced second-order finite volume scheme. The scheme is well-balanced for general equation of state, provided we can write the hydrostatic solution as a function of the space variables. This is achieved by combining hydrostatic reconstruction with contact preserving, consistent numerical flux, and appropriate source discretization. Several numerical experiments are presented to demonstrate the well-balanced property and resulting accuracy of the proposed scheme. PubDate: 2017-12-14 DOI: 10.1007/s00033-017-0901-x Issue No:Vol. 69, No. 1 (2017)

Authors:G. A. Afrouzi; M. Mirzapour; Vicenţiu D. Rădulescu Abstract: This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schrödinger-type equations of the form $$\begin{aligned} \left\{ \begin{array}{ll} - \displaystyle \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x) u ^{P^+_+-2}u =\lambda f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array}\right. \end{aligned}$$ without the Ambrosetti–Rabinowitz growth condition. Our arguments rely on the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter. PubDate: 2017-12-14 DOI: 10.1007/s00033-017-0900-y Issue No:Vol. 69, No. 1 (2017)

Authors:I. V. Andrianov; J. Kaplunov; A. K. Kudaibergenov; L. I. Manevitch Abstract: The plane strain problem for a thin circular cylindrical shell is considered within the framework of the Sanders–Koiter theory. The relative shell thickness and displacement amplitude are chosen to be of the same asymptotic order. The leading nonlinear correction to the lowest cut-off frequencies is derived using the method of multiple scales. In contrast to the traditional two-mode Galerkin expansions assuming inextensibility of the shell transverse cross section, the developed fourth-order asymptotic scheme operates with five angular modes. The obtained results reveal asymptotic inconsistency of previous approximate solutions to the problem. PubDate: 2017-12-12 DOI: 10.1007/s00033-017-0902-9 Issue No:Vol. 69, No. 1 (2017)

Authors:Xu Wang; Liang Chen; Peter Schiavone Abstract: We employ conformal mapping techniques to study the existence of internal uniform hydrostatic stresses inside two non-elliptical inclusions when the surrounding matrix is simultaneously subjected to a concentrated couple and remote uniform in-plane stresses. The unknown complex coefficients appearing in the corresponding mapping function can be determined analytically for a given pair of loading, one material and three geometric parameters. This allows us to subsequently identify the shapes of the two inclusions. Our analysis further reveals that the shapes of the inclusions depend on the concentrated couple, whereas the corresponding internal uniform hydrostatic stresses do not. PubDate: 2017-12-11 DOI: 10.1007/s00033-017-0899-0 Issue No:Vol. 69, No. 1 (2017)

Authors:Taklit Hamadouche; Salim A. Messaoudi Abstract: In this paper, we consider a linear thermoelastic Timoshenko system with variable physical parameters, where the heat conduction is given by Cattaneo’s law and the coupling is via the displacement equation. We discuss the well-posedness and the regularity of solution using the semigroup theory. Moreover, we establish the exponential decay result provided that the stability function \(\chi _{r}(x)=0\) . Otherwise, we show that the solution decays polynomially. PubDate: 2017-12-11 DOI: 10.1007/s00033-017-0897-2 Issue No:Vol. 69, No. 1 (2017)

Authors:I. Fabrikant; E. Karapetian; S. V. Kalinin Abstract: We consider the problem of an arbitrary shaped rigid punch pressed against the boundary of a transversely isotropic half-space and interacting with an arbitrary flat crack or inclusion, located in the plane parallel to the boundary. The set of governing integral equations is derived for the most general conditions, namely the presence of both normal and tangential stresses under the punch, as well as general loading of the crack faces. In order to verify correctness of the derivations, two different methods were used to obtain governing integral equations: generalized method of images and utilization of the reciprocal theorem. Both methods gave the same results. Axisymmetric coaxial case of interaction between a rigid inclusion and a flat circular punch both centered along the z-axis is considered as an illustrative example. Most of the final results are presented in terms of elementary functions. PubDate: 2017-12-09 DOI: 10.1007/s00033-017-0894-5 Issue No:Vol. 69, No. 1 (2017)