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The present paper is concerned with the existence of response solutions of quasi-periodic type for a class of quasi-periodically forced, non-Hamiltonian but reversible nonlinear beam equations. We do not suppose the basic frequency ω ∈ R 2 of the forcing term is Diophantine or Brjuno, and it might be Liouvillean, which is weaker than the Diophantine or Brjuno frequency. The proof is based on an improved Kolmogorov–Arnold–Moser (KAM) theorem for infinite-dimensional reversible systems with non-reducible normal form. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions. [ABSTRACT FROM AUTHOR]

This paper considers the problem of the local existence for the generalized MHD equations with fractional dissipative terms Λ 2 α u for the velocity field and Λ 2 β b for the magnetic field, respectively. Based on some new commutator estimates, local existence for the generalized MHD equations is established, which recovers and improves previous results. [ABSTRACT FROM AUTHOR]

In the present paper, we prove an infinite dimensional reversible Kolmogorov-Arnold-Moser (KAM) theorem. As an application, we study the existence of KAM tori for a class of two dimensional (2D) non-Hamiltonian completely resonant beam equations with derivative nonlinearities. The Birkhoff normal form theory is also used since there are no external parameters in the equations. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Based on a new gradient entropy estimate in L log L space, we prove the global existence of a continuous solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity. A comparison principle with respect to time is used for proving uniqueness of the solution for the local problem. [ABSTRACT FROM AUTHOR]

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. [ABSTRACT FROM PUBLISHER]

Mechanical degradation of electrode materials, in the form of particle cracking and fragmentation, disintegration, fracturing, and loss in contact between current collectors and the active electrode materials, can affect or deteriorate the performance of Li-ion batteries dramatically and even lead to the battery failure in electric vehicle. This paper firstly built a single particle model (SPM) based upon kinetics of electrochemical reactions. Then the Li-ion concentration, evolution of diffusion induced stresses within the SPM under potentiostatic or galvanostatic operating conditions were analyzed by utilizing a mathematical method. Next, evolution of stresses or strains in the SPM, at the core of relates with mechanical degradation of electrode materials, are elaborated in detail. Finally, surface and morphology of the electrodes dismantled from fresh and degraded cells after galvanostatic charge/discharge cycling have been analyzed to verify the hypothesis aforementioned by observing scanning electron microscopy and analyzing X-ray diffraction. [ABSTRACT FROM AUTHOR]

This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator. [ABSTRACT FROM AUTHOR]

This paper deals with the Eringen's theory for binary mixtures between elastic micropolar solids and incompressible micropolar fluids (Eringen in J Appl Phys 94:4184-4190, ). Using the weighted energy method, an uniqueness result in the case of unbounded domains for small displacement of the solid and for non-slow flow of fluid is presented. [ABSTRACT FROM AUTHOR]

In this paper, we show that there are almost periodic solutions corresponding to full dimensional invariant tori for higher dimensional Schrödinger equations with Fourier multiplier i u t −Δ u+ M ξ u+ f(| u|2) u = 0, subject to periodic boundary conditions, where the nonlinearity f is a real-analytic function near u = 0 with f(0) = 0. The proof is based on an improved infinite dimensional KAM theorem. [ABSTRACT FROM AUTHOR]

In this paper, double dissimilar orthotropic composite materials interfacial crack is studied by constructing new stress functions and employing the method of composite material complex. When the characteristic equations’ discriminants Δ1 > 0 and Δ2 > 0, the theoretical formula of the stress field and the displacement field near the mode I interface crack tip are derived, indicating that there is no oscillation and interembedding between the interfaces of the crack. [ABSTRACT FROM AUTHOR]

Weak solution (or generalized solution) for the boundary-value problems of partial differential equations of elasticity of 3D (three-dimensional) quasicrystals is given, in which the matrix expression is used. In terms of Korn inequality and theory of function space, we prove the uniqueness of the weak solution. This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals, and develops the weak solution theory of elasticity of 2D quasicrystals given by the second author of the paper and his students. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution. We also discuss an asymptotic expansion of high order in a small parameter of a weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the cauchy problem for the 2D Groma-Balogh model (Acta Mater 47:3647-3654, 1999). From the works Cannone et al. (Arch Ration Mech Anal 196:71-96, 2010) and El Hajj (Ann Inst Henri Poincaré Anal Nonlinéaire 27:21-35, 2010), one can see global well-posedness for this model is an open question. However, we can prove longtime well-posedness. In particular, we show that this model admits a unique solution with the lifespan $$T^\star $$ satisfying $$T^\star \log ^2(1+T^\star )\gtrsim \epsilon ^{-2}$$ if the initial data is of size $$\epsilon $$ . To achieve this, we first establish some new decay estimates concerning the operator $$e^{-{\mathcal {R}}_{12}^2t}$$ . Then, we prove the longtime well-posedness by utilizing the weak dissipation to deal with the nonlinear terms. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of regularity criteria for the 2D generalized MHD system with fractional dissipative terms $${-\Lambda^{2\alpha}u}$$ for the velocity field and $${-\Lambda^{2\beta}b}$$ for the magnetic field respectively. Various regularity criteria are established to guarantee smoothness of solutions. It turns out that our regularity criteria imply previous global existence results naturally. [ABSTRACT FROM AUTHOR]

In this paper we discuss the well-posedness and the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a Kelvin-Voigt string-beam system which models a suspension bridge. For this model we obtain the existence and uniqueness of solutions and the exponential stability of the homogeneous system, provided that the constant axial force $$p$$ is smaller than a critical value. For a general $$p,$$ the existence of the regular global attractor is proved when the external loads are independent of time. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear degenerate parabolic system with nonlocal source and nonlocal boundaries. By super-solution, sub-solution and auxiliary functions, a criteria for nonnegative solution of global existence and blow-up in finite time is obtained for this degenerate nonlocal problem. Finally, the blow-up rates of blow-up solutions are also estimated. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of global C solutions of the Goursat problem for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of Lipschitz continuous and piecewise C traveling wave solutions, provided that the C norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small. Applications include the 1D compressible Euler equations for Chaplygin gases. [ABSTRACT FROM AUTHOR]

The artificial boundary method is one of the most popular and effective numerical methods for solving partial differential equations on unbounded domains, with more than thirty years development. The artificial boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method. [ABSTRACT FROM AUTHOR]

This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Γ-convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance. [ABSTRACT FROM AUTHOR]

Large-scale simulations of coupled flow in deformable porous media require iterative methods for solving the systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in realistic geological applications, such as basin evolution at regional scales. The success of iterative methods for such problems depends strongly on finding effective preconditioners with good parallel scaling properties, which is the topic of the present paper. We present a parallel preconditioner for Biot's equations of coupled elasticity and fluid flow in porous media. The preconditioner is based on an approximation of the exact inverse of the two-by-two block system arising from a finite element discretisation. The approximation relies on a highly scalable approximation of the global Schur complement of the coefficient matrix, combined with generally available state-of-the-art multilevel preconditioners for the individual blocks. This preconditioner is shown to be robust on problems with highly heterogeneous material parameters. We investigate the weak and strong parallel scaling of this preconditioner on up to 512 processors and demonstrate its ability on a realistic basin-scale problem in poroelasticity with over eight million tetrahedral elements. [ABSTRACT FROM AUTHOR]

This paper establishes the global existence of classical solution to the system of homogeneous, isotropic hyperelasticity with time-independent external force, provided that the nonlinear term obeys a type of null condition. The authors first prove the existence and uniqueness of the stationary solution. Then they show that the solution to the dynamical system converges to the stationary solution as time goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Kirchhoff wave equation where $\widetilde{u}_{0}$ , $\widetilde{u}_{1}$ , μ, f, g are given functions and $\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.$ To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo-Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1) is studied. [ABSTRACT FROM AUTHOR]

In this paper, we show that every solution of the Khokhlov-Zabolotskaya equation possesses a Minkowski metric. This metric is responsible for transport of singularities of KZ-solutions. We find explicit solutions for which the metrics are either locally-flat or Ricci-flat or conformally-flat. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the following nonlinear wave equation where , , , are given functions and , are given constants. In Eq. , the nonlinear term depends on the integrals , . In this paper we associate with Eq. a recurrent sequence defined by , , with satisfying . The first term is chosen as . If and , we prove that the sequence converges at a rate of order to a unique weak solution of problem (1). [Copyright &y& Elsevier]

The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C 1 solution to Cauchy problem. [ABSTRACT FROM AUTHOR]

This article defines and presents the mathematical analysis of a new class of models from the theory of inelastic deformations of metals. This new class, containing so called convex composite models, enlarges the class containing monotone models of gradient type defined in [1]. This paper starts to establish the existence theory for models from this new class; we restrict our investigations to the coercive and linear self-controlling case. [ABSTRACT FROM AUTHOR]

In linear elasticity problems, the pressure is usually introduced for computing the incompressible state. In this paper is presented a technique which is based on a power series expansion of the displacement with respect to the inverse of Lamé's coefficient $\lambda$ . It does not require to introduce the pressure as an auxiliary unknown. Moreover, low degree finite elements can be used. The same technique can be applied to Stokes or Navier-Stokes equations, and can be extended to more general parameterized partial differential equations. Discretization error and convergence are analyzed and illustrated by some numerical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with 'non-constant coefficients' and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with 'constant coefficients' and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method. [ABSTRACT FROM AUTHOR]

A method for determining the amplitude-dependent mode shapes and the corresponding modal dynamics of weakly nonlinear vibratory systems is described. The method is a combination of a Galerkin projection and invariant manifold techniques and is applied to a class of distributed parameter vibratory systems. In this paper the general theory for a class of conservative systems is outlined and applied to determine the nonlinear mode shapes and modal dynamics of a linear Euler-Bernoulli team attached to a nonlinear elastic foundation. [ABSTRACT FROM AUTHOR]

The thermoelasticity theories describing a finite speed for the propagation of thermal perturbation have been investigated over the last 50 years. In contrast to the classical thermoelasticity theory, these nonclassical theories involve a hyperbolic type heat transport equation and they are motivated by experiments exhibiting the actual occurence of wave‐type heat transport, so‐called second sound (cf. [6–8,11,26]). The aim of this paper is to present a new system of equations describing nonlocal model of thermoelasticity theory. In order to obtain it we used the Pipkin and Gurtin [26] approach to the constitutive relations for the internal energy and the heat flux. Using the modified Cagniard‐de Hoop method we constructed the matrix of fundamental solutions to this system of equations in the three‐dimensional spaces. Basing on the constructed matrix of fundamental solutions we obtained the explicit formulae for the solution to the Cauchy problem to this system of equations in the form of some kind of convolutions. Applying the method of Sobolev spaces we proved the Lp−Lq time decay estimate for the solution to the considered Cauchy problem to this new system of equations. We are interested in physical and mathematical aspects of this new system of equations. We discovered the new type of thermal wave appearing in the medium described by the nonlocal model of thermoelasticity theory. [ABSTRACT FROM AUTHOR]

In this article, we apply blow-up analysis to study pointwise a priori estimates for some p-Laplacian equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function, i.e., the solution of Δpu = 0, x ∈ Ω. We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplacian equation −Δpu = |u|λ − 1u, x ∈ Ω, which are then extended to the equation with general right hand term f(x, u) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplacian equation, in a case of p = 2, are also discussed. [ABSTRACT FROM AUTHOR]

An initial-boundary value problem of Saut–Temam type for the modified Zakharov–Kuznetsov equation posed on a strip is considered. Critical power in nonlinearity has been studied. The results on existence, uniqueness and asymptotic behavior of solutions are presented. [ABSTRACT FROM AUTHOR]

We study the energy decay of the solutions of a linear homogeneous anisotropic porous thermoelastic system in the context of Green and Naghdi model of type II with the following boundary condition with memory for the displacement $${{\bf T}(x,t)n(x) = -\gamma_0v(x,t) - \int_0^\infty \lambda(s)v^t(x,s) {{d}}s}$$ . By introducing a boundary free energy, we prove that if the kernel λ exponentially decays in time, then also the energy exponentially decays when porosity viscosity is present. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term . First, we establish an upper bound for the difference between the solution of our equation and the heat equation in space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method. [Copyright &y& Elsevier]

Abstract: In this paper we prove local well-posedness in and for the generalized sixth-order Boussinesq equation . Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) . We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of solutions are obtained for small initial and boundary data. We also present two applications for physical models. [Copyright &y& Elsevier]

Abstract: We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a parabolic equation with a non-local term defined on a compact two-dimensional Riemannian surface . If the total mass of the solution, , is equal to and is the standard sphere , it is a Hamilton’s normalized Ricci flow. We obtain the global in time existence of the solution to this problem for . If , the orbit is compact while for , there is a time sequence along which the solution converges to a stationary solution. [Copyright &y& Elsevier]

Abstract: In this paper, we study the model of Groma and Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show that this model has a unique local-in-time solution corresponding to any initial datum in the space for and , where is the Hölder–Zygmund space. [Copyright &y& Elsevier]