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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]

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]

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]

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]

Initial-boundary value problems for nonlinear dispersive equations of evolution of order 2 l + 1 , l ∈ N with a convective term of the form u k u x , k ∈ N have been considered on intervals (0 , L) , L ∈ (0 , + ∞) . Definitions of regular and critical values of k as functions of l have been given. The existence and uniqueness of global regular solutions as well as exponential decay of them for small initial data have been established. [ABSTRACT FROM AUTHOR]

We present a robust optimization framework that is applicable to general nonlinear programs (NLP) with uncertain parameters. We focus on design problems with partial differential equations (PDE), which involve high computational cost. Our framework addresses the uncertainty with a deterministic worst-case approach. Since the resulting min-max problem is computationally intractable, we propose an approximate robust formulation that employs quadratic models of the involved functions that can be handled efficiently with standard NLP solvers. We outline numerical methods to build the quadratic models, compute their derivatives, and deal with high-dimensional uncertainties. We apply the presented approach to the parametrized shape optimization of systems that are governed by different kinds of PDE and present numerical results. [ABSTRACT FROM AUTHOR]

In this article, we study a globally modified Allen-Cahn-Navier-Stokes system in a three-dimensional domain. The model consists of the globally modified Navier-Stokes equations proposed in Caraballo et al. (Adv Nonlinear Stud 6(3):411-436, 2006) for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. We prove the existence and uniqueness of strong solutions. Using the flattening property, we also prove the existence of global V-attractors for the model. Using a limiting argument, we derive the existence of bounded entire weak solutions for the three-dimensional coupled Allen-Cahn-Navier-Stokes system with time-independent forcing. [ABSTRACT FROM AUTHOR]

In this article, we study a globally modified Cahn–Hilliard–Navier–Stokes system in a three-dimensional domain. The model consists of the globally modified Navier–Stokes equations proposed, coupled with a Cahn–Hilliard model. We prove the existence and uniqueness of strong solutions. Using the flattening property, we also prove the existence of globalattractors for the model. Using a limiting argument, we derive the existence of bounded entire weak solutions for the three-dimensional coupled Cahn–Hilliard–Navier–Stokes system with time-independent forcing. [ABSTRACT FROM PUBLISHER]

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle. [ABSTRACT FROM PUBLISHER]

We study a dynamical thin shallow shell whose elastic deformations are described by a nonlinear system of Marguerre-Vlasov's type under the presence of thermal effects. Our main result is the proof of a global existence and uniqueness of a weak solution in the case of clamped boundary conditions. Standard techniques for uniqueness do not work directly in this case. We overcame this difficulty using recent work due to Lasiecka (Appl Anal 4:1376-1422, 1998). [ABSTRACT FROM AUTHOR]

The fracture behaviors near the mode II interface crack tip for orthotropic bimaterial are studied. The non-oscillatory field, where the stress singularity exponent is a real number, is discussed by the complex function method and the undetermined coefficient method. From the research fracture problems, the stress functions with ten undetermined coefficients and an unknown singularity exponent are introduced when Δ > 0 and Δ > 0. By the existence theorem of non-trival solutions for the system of eight homogeneous linear equations, the characteristic equation, the stress singularity exponent, and the discriminating condition of the non-oscillatory singularity are found. By the uniqueness theorem of the solutions for the system of twelve non-homogeneous linear equations with ten unknowns, the ten undermined coefficients in the stress functions are uniquely determined. The definitions of the stress intensity factors are given with the help of one-sided limit, and their theoretical formulae are deduced. The analytic solutions of the stresses near the mode II interface crack tip are derived. The classical results for orthotropic material are obtained. [ABSTRACT FROM AUTHOR]

Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional (3D) icosahedral quasicrystals are discussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhesive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regulation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason coupling constant impact is barely perceptible. The validity of the conclusions is verified. [ABSTRACT FROM AUTHOR]

We revisit the time-incremental method for proving existence of a quasistatic evolution in perfect plasticity. We show how, as a consequence of a priori time regularity estimates on the stress and the plastic strain, the piecewise affine interpolants of the solutions of the incremental minimum problems satisfy the conditions defining a quasistatic evolution up to some vanishing error. This allows for a quicker proof of existence: furthermore, this proof bypasses the usual variational reformulation of the problem and directly tackles its original mechanical formulation in terms of an equilibrium condition, a stress constraint, and the principle of maximum plastic work. [ABSTRACT FROM AUTHOR]