Second-order Evolution Equations Research Articles

Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback

We consider a stabilization problem for abstract second-order evolution equations with dynamic boundary feedback laws with a delay and distributed structural damping. We prove an exponential stability result under a suitable condition between the internal damping and the boundary laws. The proof of the main result is based on an identity with multipliers that allows to obtain a uniform decay estimate for a suitable energy functional. Some concrete examples are detailed. Some counterexamples suggest that this condition is optimal.

Almost periodic solutions to some second-order nonautonomous differential equations

This paper is concerned with the existence of almost periodic mild solutions to some second-order equations. Using dichotomy tools and the Schauder fixed point theorem, the existence of almost periodic mild solutions to those second-order evolution equations is established. To illustrate our abstract results, the existence of almost periodic solutions to a damped second-order boundary value problem is also discussed.

Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization. The discretization combines a time stepping on a non-uniform time grid, which generalizes the well-known Störmer–Verlet scheme, with an internal approximation scheme. The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Hölder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces. Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.

Full discretisation of second-order nonlinear evolution equations: strong convergence and applications

Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

Exponential Stability of Two Coupled Second-Order Evolution Equations

By using the multiplier technique, we prove that the energy of a system of two coupled second order evolution equations (one is an integro-differential equation) decays exponentially if the convolution kernel decays exponentially. An example is give to illustrate that the result obtained can be applied to concrete partial differential equations.

A nonlinear second-order evolution equation – classical and nonclassical symmetry analyses and their physical implications

A nonlinear second-order evolution equation – classical and nonclassical symmetry analyses and their physical implications

Energy decay rates via convexity for some second-order evolution equation with memory and nonlinear time-dependent dissipation

The stabilization of the following abstract integro-differential equation: u ″ ( t ) + A u ( t ) + ∫ 0 t g ( t − s ) A u ( s ) d s + Q ( t , u ′ ( t ) ) = ∇ F ( u ( t ) ) , is investigated. We establish the general decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g , without imposing any restrictive growth assumption on the damping at the origin and strongly weakening the usual assumption of the relaxation function g . Our approach is based on the multiplier method and make use of some properties of the convex functions. These decay results can be applied to various concrete models. We shall study some examples to illustrate our result.

Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations ¨(t )+ γ ˙ u(t )+ ∇φ(u(t)) + A(u(t)) = 0, where ∇φ is the gradient operator of a convex differentiable potential function φ : H→ R, A : H→H is a maximal monotone operator which is assumed to be λ-cocoercive, and γ> 0i s a damping parameter. Potential and non-potential effects are associated respectively to ∇φ and A. Under condition λγ 2 > 1, it is proved that each trajectory asymptotically weakly converges to a zero of ∇φ + A. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.

Second-order Neutral Stochastic Evolution Equations with Infinite Delay under Carathéodory Conditions

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay, in which the initial value belongs to the abstract space ℬ. We establish the existence and uniqueness of mild solutions for SNSEEIs under global and local Caratheodory conditions by means of the successive approximation. An application to the stochastic nonlinear wave equations with infinite delay is given to illustrate the theory.

Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order

The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfils a certain growth and a Holder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous operator that fulfils a certain growth condition and is (up to some shift) monotone and coercive.

Nonlocal symmetries of evolution equations

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras of the dimension up to three. As a result, we construct the broad families of new nonlinear evolution equations possessing nonlocal symmetries which in principle cannot be obtained within the classical Lie approach.

Practical tools for third order cosmological perturbations

We discuss cosmological perturbation theory at third order, derivingthe gauge transformation rules for metric and matter perturbations,and constructing third order gauge invariant quantities. We presentthe Einstein tensor components, the evolution equations for a perfectfluid, and the Klein-Gordon equation at third order, including scalar,vector and tensor perturbations. In doing so, we also give all secondorder tensor components and evolution equations in full generality.

Second-order neutral impulsive stochastic evolution equations with delay

In this paper, we study the second-order neutral stochastic evolution equations with impulsive effect and delay (SNSEEIDs). We establish the existence and uniqueness of mild solutions to SNSEEIDs under non-Lipschitz condition with Lipschitz condition being considered as a special case by the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial data by means of corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with impulsive effect and delay is given to illustrate the theory.

On relation between potential and contact symmetries of evolution equations

We prove that any evolution equation, that can be written in the form of a conservation law, reduces to an evolution equation admitting the group of contact symmetries leaving the temporal variable invariant and vice versa. One of the consequences of this fact is that any evolution equation admitting the potential symmetries can be transformed into another evolution equation so that the potential symmetries map into contact symmetries of the latter. Based on this fact is out group approach to classification of evolution equations possessing nonlocal symmetries. We present several examples of classifications of second-order evolution equations admitting potential symmetries.

Constraining relativistic viscous hydrodynamical evolution

We show that by requiring positivity of the longitudinal pressure it is possible to constrain the initial conditions one can use in second-order viscous hydrodynamical simulations of ultrarelativistic heavy-ion collisions. We demonstrate this explicitly for $(0+1)$-dimensional viscous hydrodynamics and discuss how the constraint extends to higher dimensions. Additionally, we present an analytic approximation to the solution of $(0+1)$-dimensional second-order viscous hydrodynamical evolution equations appropriate to describe the evolution of matter in an ultrarelativistic heavy-ion collision.

A note on the dimension of the global attractor for an abstract semilinear hyperbolic problem

We study a semilinear hyperbolic problem, written as a second-order evolution equation in an infinite-dimensional Hilbert space. Assuming existence of the global attractor, we estimate its fractal dimension explicitly in terms of the data. Despite its elementary character, our technique gives reasonable results. Notably, we require no additional regularity, although nonlinear damping is allowed.

Second-order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls

In this article, the second-order nonlinear impulsive evolution differential equations with time-varying generating operators is considered. Constructing evolution systems generated by time-varying operator matrix, we introduce suitable mild solution of the second-order nonlinear impulsive evolution differential equations. The existence and uniqueness of the mild solutions and the continuous dependence on initial value are proved. The existence of the optimal controls for a Lagrange problem of the systems governed by the second-order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.

Almost automorphic solutions to second-order semilinear evolution equations

In this paper we give some sufficient conditions for ensuring the existence and uniqueness of a mild almost automorphic solution to a second-order semilinear evolution equation in a Banach space. We also present some properties of the (new) notion of a uniform spectrum of bounded functions.

Asymptotic behavior of bounded solutions to a nonhomogeneous second-order evolution equation of monotone type

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale university, 1981, part I, pp. 1–76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465–476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226–235] for dissipative systems, we study the asymptotic behavior of solutions to the following system of second-order nonhomogeneous evolution equations: { u ″ ( t ) ∈ A u ( t ) + f ( t ) a.e. t ∈ ( 0 , + ∞ ) u ( 0 ) = u 0 , sup t ≥ 0 | u ( t ) | < + ∞ where A is a maximal monotone operator in a real Hilbert space H , and f : R + → H is a suitable function. We investigate weak and strong convergence theorems for solutions to this system. Our results extend previous results of Morosanu [G. Morosanu, Nonlinear Evolution Equations and Applications, Editura Academiei Romane (and D. Reidel publishing Company, Bucharest, 1988; G. Morosanu, Asymptotic behaviour of solutions of differential equations associated to monotone operators, Nonlinear Anal. 3 (1979) 873–883] and Mitidieri [E. Mitidieri, Some remarks on the asymptotic behaviour of the solutions of second order evolution equations, J. Math. Anal. Appl. 107 (1985) 211–221] who studied the case f ≡ 0 by assuming that A − 1 ( 0 ) ≠ ϕ , as well as previous results of the authors [B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math. (in press)] who studied the homogeneous case without this additional assumption.

Semilinear Evolution Equations of Second Order via Maximal Regularity

Semilinear Evolution Equations of Second Order via Maximal Regularity