Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 2011"

Plane wave approximation of homogeneous Helmholtz solutions

Abstract: In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω2u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.

Existence and asymptotic stability of a viscoelastic wave equation with a delay

Abstract: In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problem $$u_{tt}(x,t)-\Delta u(x,t)+\int\limits_{0}^{t}g(t-s){\Delta}u(x,s){d}s+\mu_{1}u_{t}(x,t)+\mu_{2} u_{t}(x,t-\tau)=0$$ together with initial conditions and boundary conditions of Dirichlet type. Here $${(x,t)\in\Omega\times (0,\infty), g}$$ is a positive real valued decreasing function and μ 1, μ 2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo–Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals.

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

Abstract: In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrodinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.

Hopf bifurcation for non-densely defined Cauchy problems

Abstract: In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.

Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations

Abstract: This paper is concerned with the multiplicity and concentration of positive solutions for the nonlinear Schrodinger–Poisson equations $$ \left\{ \begin{array}{l@{\quad}l} -\varepsilon^2\triangle u+V(x)u+\phi(x) u=f(u)& {\rm in}\,{\mathbb R}^3, \\ -\varepsilon^2\triangle \phi=u^2 & {\rm in}\,{\mathbb R}^3, \\ u\in H^1({\mathbb R}^3), u(x) > 0,& \forall x\in{\mathbb R}^3, \\ \end{array} \right. $$ where e > 0 is a parameter, $${V: {\mathbb R}^3\rightarrow{\mathbb R}}$$ is a continuous function and $${f: {\mathbb R}\rightarrow {\mathbb R}}$$ is a C 1 function having subcritical growth. The proof of the main result is based on minimax theorems and the Ljusternik–Schnirelmann theory.

Stability of anisotropic electroactive polymers with application to layered media

Abstract: The stability of anisotropic electroactive polymers is investigated. A general criterion for the onset of instabilities under plane-strain conditions is introduced in terms of a sextic polynomial whose coefficients depend on the instantaneous electroelastic moduli. In a way of an example, the stable domains of layered neo-Hookean dielectrics are determined. It is found that depending on the direction of the electrostatic excitation field relative to the lamination direction, the critical stretch ratios at which instabilities may occur can be either larger or smaller than the ones for the purely mechanical case.

Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures

Abstract: A new mathematical model of two-temperature magneto-thermoelasticity is constructed where the fractional order heat conduction law is considered. The state space approach is adopted for the solution of one-dimensional application for a perfect conducting half-space of elastic material with heat sources distribution in the presence of a transverse magnetic field. The Laplace-transform technique is used. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusions about the new theory are given. Some comparisons are shown in figures to estimate the effects of the temperature discrepancy and the fractional order parameter on all the studied fields.

Stable standing waves for a class of nonlinear Schrödinger-Poisson equations

Abstract: We prove the existence of orbitally stable standing waves with prescribed L2-norm for the following Schrodinger-Poisson type equation $$i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 \quad \rm{in} \quad \mathbb R^{3},$$ when \({p\in \left\{ \frac{8}{3}\right\}\cup (3,\frac{10}{3})}\). In the case \({3 < p < \frac{10}{3}}\), we prove the existence and stability only for sufficiently large L2-norm. In case \({p=\frac{8}{3}}\), our approach recovers the result of Sanchez and Soler (J Stat Phys 114:179–204, 2004) for sufficiently small charges. The main point is the analysis of the compactness of minimizing sequences for the related constrained minimization problem. In the final section, a further application to the Schrodinger equation involving the biharmonic operator is given.

A thermodynamical framework for chemically reacting systems

Abstract: In this paper, we develop a thermodynamic framework that is capable of describing the response of viscoelastic materials that are undergoing chemical reactions that takes into account stoichiometry. Of course, as a special sub-case, we can also describe the response of elastic materials that undergo chemical reactions. The study generalizes the framework developed by Rajagopal and co-workers to study the response of a disparate class of bodies undergoing entropy producing processes. One of the quintessential feature of this framework is that the second law of thermodynamics is formulated by introducing Gibbs’ potential, which is the natural way to study problems involving chemical reactions. The Gibbs potential–based formulation also naturally leads to implicit constitutive equations for the stress tensor. Another feature of the framework is that the constraints due to stoichiometry can also be taken into account in a consistent manner. The assumption of maximization of the rate of entropy production due to dissipation, heat conduction, and chemical reactions is invoked to determine an equation for the evolution of the natural configuration κ p(t)(B), the heat flux vector and a novel set of equations for the evolution of the concentration of the chemical constituents. To determine the efficacy of the framework with regard to chemical reactions, those occurring during vulcanization, a challenging set of chemical reactions, are chosen. More than one type of reaction mechanism is considered and the theoretically predicted distribution of mono, di and polysulfidic cross-links agree reasonably well with available experimental data.

A model for surface diffusion of trans-membrane proteins on lipid bilayers

Abstract: The equilibrium theory of lipid membranes is modified to include the effects of a continuous distribution of trans-membrane proteins. These influence membrane shape and evolve in accordance with a diffusive balance law. The model is purely mechanical in the absence of the proteins. Conditions ensuring energy dissipation in the presence of diffusion are given and an example constitutive function is used to simulate the coupled inertia-less interplay between membrane shape and protein distribution. The work extends an earlier continuum theory of equilibrium configurations of composite lipid-protein membranes to accommodate surface diffusion.

On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite

Abstract: The stress field inside a two-dimensional arbitrary-shape elastic inclusion bonded through an interphase layer to an infinite elastic matrix subjected to uniform stresses at infinity is analytically studied using the complex variable method in elasticity. Both in-plane and anti-plane shear loading cases are considered. It is shown that the stress field within the inclusion can be uniform and hydrostatic under remote constant in-plane stresses and can be uniform under remote constant anti-plane shear stresses. Both of these uniform stress states can be achieved when the shape of the inclusion, the elastic properties of each phase, and the thickness of the interphase layer are properly designed. Possible non-elliptical shapes of inclusions with uniform hydrostatic stresses induced by in-plane loading are identified and divided into three groups. For each group, two conditions that ensure a uniform hydrostatic stress state are obtained. One condition relates the thickness of the interphase layer to elastic properties of the composite phases, while the other links the remote stresses to geometrical and material parameters of the three-phase composite. Similar conditions are analytically obtained for enabling a uniform stress state inside an arbitrary-shape inclusion in a three-phase composite loaded by remote uniform anti-plane shear stresses.

Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems

Abstract: We extend the study of the integrability done by Leach and Miritzis (J Nonlinear Math Phys 13:535–548, 2006) on the classical model of competition between three species studied by May and Leonard (SIAM J Appl Math 29:243–256, 1975), to all real values of the parameters. Additionally, our results provide all polynomial, rational and analytic first integrals of this extended model. We also classify all the invariant algebraic surfaces of these models.

Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow

Abstract: We consider a 2D system that models the nematic liquid crystal flow through the Navier–Stokes equations suitably coupled with a transport-reaction-diffusion equation for the averaged molecular orientations. This system has been proposed as a reasonable approximation of the well-known Ericksen–Leslie system. Taking advantage of previous well-posedness results and proving suitable dissipative estimates, here we show that the system endowed with periodic boundary conditions is a dissipative dynamical system with a smooth global attractor of finite fractal dimension.

Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

Abstract: This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossing-monostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson’s blowflies equation in population dynamics and Mackey–Glass model in physiology.

Multiple solutions for a superlinear and periodic elliptic system on {\mathbb{R}^N}

Abstract: This paper is concerned with the following periodic Hamiltonian elliptic system $$\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.$$ where the potential V is periodic and 0 lies in a gap of the spectrum of −Δ + V, f(x, t) and g(x, t) depend periodically on x and are superlinear but subcritical in t at infinity. By establishing a variational setting, existence of a ground state solution and multiple solution for odd f and g are obtained.

Stokes flow of an assemblage of porous particles: stress jump condition

Abstract: The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (β), volume fraction (e), and effective viscosity.

Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x)

Abstract: We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function r+2Fr+1(x) when there are additional numeratorial and denominatorial parameters differing by unity. The method employed to deduce the latter is also implemented to obtain a Kummer-type transformation formula for r+1Fr+1 (x) that was recently derived in a different way.

Vekua theory for the Helmholtz operator

Abstract: Vekua operators map harmonic functions defined on domain in \({\mathbb R^{2}}\) to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

Global solutions in higher dimensions to a fourth-order parabolic equation modeling epitaxial thin-film growth

Abstract: The initial-value problem for $$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta | abla u|^2 + f(x)\qquad \qquad (\star)$$ is studied under the conditions $${{\frac{\partial}{\partial u}} u={\frac{\partial}{\partial u}} \Delta u=0}$$ on the boundary of a bounded convex domain $${\Omega \subset {\mathbb{R}}^n}$$ with smooth boundary. This problem arises in the modeling of the evolution of a thin surface when exposed to molecular beam epitaxy. Correspondingly the physically most relevant spatial setting is obtained when n = 2, but previous mathematical results appear to concentrate on the case n = 1. In this work, it is proved that when n ≤ 3, μ ≥ 0, λ > 0 and $${f \in L^\infty(\Omega)}$$ satisfies $${{\int_\Omega} f \ge 0}$$ , for each prescribed initial distribution $${u_0 \in L^\infty(\Omega)}$$ fulfilling $${{\int_\Omega} u_0 \ge 0}$$ , there exists at least one global weak solution $${u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}$$ satisfying $${{\int_\Omega} u(\cdot,t) \ge 0}$$ for a.e. t > 0, and moreover, it is shown that this solution can be obtained through a Rothe-type approximation scheme. Furthermore, under an additional smallness condition on μ and $${\|f\|_{L^\infty(\Omega)}}$$ , it is shown that there exists a bounded set $${S\subset L^1(\Omega)}$$ which is absorbing for $${(\star)}$$ in the sense that for any such solution, we can pick T > 0 such that $${e^{2\lambda u(\cdot,t)}\in S}$$ for all t > T, provided that Ω is a ball and u 0 and f are radially symmetric with respect to x = 0. This partially extends similar absorption results known in the spatially one-dimensional case. The techniques applied to derive appropriate compactness properties via a priori estimates include straightforward testing procedures which lead to integral inequalities involving, for instance, the functional $${{\int_\Omega} e^{2\lambda u}dx}$$ , but also the use of a maximum principle for second-order elliptic equations.

Micromorphic continuum model for electromagnetoelastic solids

Abstract: The microcontinuum theory of electroelasticity is considered for polarizable dielectrics on the basis of dipole and quadrupole densities as microfields. Electromagnetic contributions to force, couple, and power are derived, and their correspondence with quantities evaluated in terms of macroscopic polarization and magnetization is examined. A constitutive model that accounts for dissipation is proposed via internal variables satisfying suitable evolution equations. This approach reveals different roles of polarization and strain measures in dissipative processes. The link between the spin inertia tensor and the pair of dipole and quadrupole per unit mass is exploited to derive a nonlinear system of governing equations for a reduced set of variables. The special cases of microstretch and micropolar continua are discussed.

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

Abstract: The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.

Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

Abstract: This work is concerned with a system of nonlinear viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We will prove that the energy associated to the system is unbounded. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. The key ingredient in the proof is a method used in Vitillaro (Arch Ration Mech Anal 149:155–182, 1999) and developed in Said-Houari (Diff Integr Equ 23(1–2):79–92, 2010) for a system of wave equations, with necessary modification imposed by the nature of our problem.

On Gaussian decay estimates of solutions to some linear elliptic equations and their applications

Abstract: In this paper we establish pointwise decay estimates of solutions to some linear elliptic equations by using the Nash–Moser iteration arguments and the ODE method. As applications we obtain sharp Gaussian decay estimates for solutions to nonlinear elliptic equations that are related with self-similar solutions to nonlinear heat equations and standing wave solutions to nonlinear Schrodinger equations with harmonic potential.

Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

Abstract: We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur.

Jump conditions in stress relaxation and creep experiments of Burgers type fluids: a study in the application of Colombeau algebra of generalized functions

Abstract: We discuss stress relaxation and creep experiments of fluids that are generalizations of the classical model due to Burgers by allowing the material moduli such as the viscosities and relaxation and retardation times to depend on the stress. The physical problem, which is cast within the context of one dimension, leads to an ordinary differential equation that involves nonlinear terms like product of a function with a jump discontinuity and the derivative of a function with a jump discontinuity. As the equations are nonlinear, standard techniques that are used to study problems concerning linear viscoelastic fluids such as Laplace transforms and the theory of distributions are not applicable. We find it necessary to seek the solution in a more general setting. We discuss the mathematical and physical issues concerning the jump discontinuities and nonlinearity of the governing equation, and we show that the solution to the governing equation can be found in the sense of the generalized functions introduced by Colombeau. In the framework of Colombeau algebra we, under certain assumptions, derive jump conditions that shall be used in stress relaxation and creep experiments of fluids of the Burgers type. We conclude the paper with a discussion of the physical relevance of these assumptions.

Radiation effects on the MHD flow near the stagnation point of a stretching sheet: revisited

Abstract: This paper considers the effects of radiation on the flow near the two-dimensional stagnation point of a stretching sheet immersed in a viscous and incompressible electrically conducting fluid in the presence of an applied constant magnetic field. The external velocity and the stretching velocity of the sheet are assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a system of ordinary differential equations using a similarity transformation, before being solved numerically by the Keller-box method. The features of the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The results indicate that the heat transfer rate at the surface decreases in the presence of radiation.

On the wave propagation in isotropic fractal media

Abstract: In this paper, we explore the wave propagation phenomenon in three-dimensional (3D) isotropic fractal media through analytical and computational means. We present the governing scalar wave equation, perform its eigenvalue decomposition, and discuss its corresponding modal solutions. The homogenization through which this fractal wave equation is derived makes its mathematical analysis and consequently the formulation of exact solutions possible if treated in the spherical coordinate system. From the computational perspective, we consider the finite element method and derive the corresponding weak formulation which can be implemented in the numerical scheme. The Newmark time-marching method solves the resulting elastodynamic system and captures the transient response. Two solvers capable of handling problems of arbitrary initial and boundary conditions for arbitrary domains are developed. They are validated in space and time, with particular problems considered on spherical shell domains. The first solver is elementary; it handles problems of purely radial dependence, effectively, 1D. However, the second one deals with general advanced 3D problems of arbitrary spatial dependence.

Uniqueness for the coagulation-fragmentation equation with strong fragmentation

Abstract: The existence of at least one mass-conserving solution for continuous coagulation-fragmentation equation has been established by Escobedo et al. (J Differ Equ 195:143–174, 2003) for a large class of coagulation kernels under strong binary fragmentation. In this work, uniqueness of mass-conserving solutions is demonstrated with some additional restrictions on the fragmentation kernels.

Incompressible laminar flow through hollow fibers: a general study by means of a two-scale approach

Abstract: We study the laminar flow of an incompressible Newtonian fluid in a hollow fiber, whose walls are porous. We write the Navier–Stokes equations for the flow in the inner channel and Darcy’s law for the flow in the fiber, coupling them by means of the Beavers–Joseph condition which accounts for the (possible) slip at the membrane surface. Then, we introduce a small parameter \({\varepsilon \ll 1}\) (the ratio between the radius and the length of the fiber) and expand all relevant quantities in powers of e. Averaging over the fiber cross section, we find the velocity profiles for the longitudinal flow and for the cross-flow, and eventually, we determine the explicit expression of the permeability of the system. This work is also preliminary to the study of more complex systems comprising a large number of identical fibers (e.g., ultrafiltration modules and dialysis).

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

Abstract: The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.