Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion.

Stochastic differential delay equations and their asymptotic behavior have received much attention in recent years (see [1, 2, 5, 7–10, 12, 13] and the references therein), since these retarded problems often appear in physics, biology, engineering, etc. The delays can enter into the formulations in very different ways, e.g., as a constant or variable delay or as a distributed one; some of them can even appear in the model at the same time. However, it is possible to consider all of them under a unified formulation by using appropriate differential functional equations. In this sense, we will carry out our analysis in a functional framework that will cover a wide variety of situations containing finite delays. There exists a vast literature concerning pathwise exponential stability of parabolic stochastic evolution equations (with and without delays). We mention here, among many others, [4, 11–13] and the references therein. In the case without hereditary characteristics, the problem of the asymptotic stability of delay stochastic partial differential equations of second order in time has been considered by Curtain in [6], where one can find sufficient conditions for the exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths, when the main operator generates a strongly continuous contraction semigroup. In this paper, we shall develop the theory in a variational framework for nonlinear operators in general. We are going to consider the following model:

In this paper, we consider the global smooth solutions and their decay for the full compressible magnetohydrodynamic equations in R 3. We prove the global existence of smooth solutions near the constant state in Sobolev norms by energy method and show the convergence rates of the L p -norm of these solutions to the constant state when the L q -norm of the perturbation is bounded.

1. Introduction. Let X be a Banach space with norm [l'l[ and B(Y, X) be the totality of bounded linear operators mapping Y into X. A subset C(t), t e R of B(X,X) is called a strongly continuous cosine family in X if (

Note on certain nonlinear evolution equations of second order

Asymptotic stability for some nonlinear evolution equations of second order with unbounded dissipative terms

Semilinear Evolution Equations of Second Order via Maximal Regularity

A condition on a family ${B(t):t ∈ [0,T]}$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing

Nonlocal symmetry and generation of solutions for reaction-diffusion equations

On stability and convergence of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.

Controllability and observability are discussed for a model of coupled bending torsional vibrations of a flexible beam. This model can be described by a second-order evolution equation on an appropriate Hilbert space. By using frequency-domain methods it is shown that controllability and observability of the flexible model can be checked by the conditions for the corresponding simplified first-order system.

General analytical solutions are obtained for the planar orientation structure of rigid ellipsoid of revolutions subjected to an arbitrary homogeneous flow in a Newtonian fluid. Both finite and infinite aspect ratio particles are considered. The orientation structure is described in terms of two-dimensional, time-dependent tensors that are commonly employed in constitutive equations for anisotropic fluids such as fiber suspensions. The effect of particle aspect ratio on the evolution of orientation structure is studied in simple shear and planar elongational flows. With the availability of analytical solutions, accuracies of quadratic closure approximations used for nonhomogeneous flows are analyzed, avoiding numerical integration of orientation distribution function. In general, fourth-order orientation evolution equations with sixth-order quadratic closure approximations yield more accurate representations compared to the commonly used second-order evolution equations with fourth-order quadratic closure approximations. However, quadratic closure approximations of any order are found to give correct maximum orientation angle (i.e., preferred direction) results for all particle aspect ratios and flow cases.

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting methods of lower order, e.g. second-order. Our aim is to develop higher-order ADI methods, which are performed by Richardson extrapolation, Crank-Nicolson methods and higher-order LOD methods, based on locally higher-order methods. We discuss the new theoretical results of the stability and consistency of the ADI methods. The main idea is to apply a higher- order time discretization and combine it with the ADI methods. We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives. The higher-order methods are unconditionally stable. Some numerical experiments verify our results. AMS subject classifications: 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50, 80A20, 80M25