Second-order Evolution Equations Research Articles

A weighted $$L_q(L_p)$$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

A weighted $$L_q(L_p)$$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

Second-order neutral impulsive stochastic evolution equations with infinite delay: existence, uniqueness and averaging principle

In this paper, a class of second-order neutral impulsive stochastic evolution equations with infinite delay driven by fractional Brownian motion and Poisson jumps is considered. We utilise the Carathéodory approximation technique, the Burkhölder–Davies–Gundy inequality and the Gronwall inequality to present the existence and uniqueness of mild solutions for the addressed system under Carathéodory conditions. Furthermore, we derive an averaging principle for the proposed system which is approximated by the simplified equation in the sense of mean square. Finally, an application is used to illustrate the obtained theoretical results.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Numerical treatment for some abstract degenerate second-order evolutionary problem

This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by θ, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.

Approximate controllability for second-order stochastic neutral evolution equations with infinite delay

Approximate controllability for second-order stochastic neutral evolution equations with infinite delay

Stability for a Coupled System of Second-Order Evolution Equations with Indirect Memory-Damping

Stability for a Coupled System of Second-Order Evolution Equations with Indirect Memory-Damping

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Existence and stability of pullback exponential attractors for a nonautonomous semilinear evolution equation of second order

We consider a nonautonomous semilinear evolution problem that models some sort of propagation problem in nonlinear elastic rods and nonlinear ion-acoustic waves. We investigate the existence and stability of a family of pullback exponential attractors for our problem under suitable growth and dissipativeness conditions. Moreover, we also prove the upper and lower semicontinuity of this family of pullback exponential attractors at time zero. As a particular case, we obtain the existence of the pullback attractor in an appropriate space, we prove its upper semicontinuity and, lastly, we obtain a regularity result of this pullback attractor.

A Lie group treatment on a generalized evolution Fisher type equation with variable coefficients

In this study, a general second-order evolution equation of the Fisher-type, with time-dependent variable coefficients, is considered. This equation contains many well-known equations, and obtained results may be applicable in investigating other evolution equations. Lie symmetries and corresponding invariant solutions of the considered problem are studied by a non-traditional Lie symmetry method (LSM). Prolongation of the model is an essential part of the Lie symmetry method, which in the current work, we analyze by the heir-equations. Finally, different types of solutions depending on the variable coefficients, such as the exponential solutions, trigonometric solutions, Bessel solutions, and Mathieu solutions, are extracted.

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

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.

Asymptotic Regularity and Existence of Time-Dependent Attractors for Second-Order Undamped Evolution Equations with Memory

Our purpose in this article is to study the asymptotic behavior of undamped evolution equations with fading memory on time-dependent spaces. By means of the theory of processes on time-dependent spaces, asymptotic a priori estimate and the technique of operator decomposition and the existence and asymptotic regularity of time-dependent attractors are, respectively, established in the critical case. At the same time, we also obtain the asymptotic regularity of the solution.

Approximate controllability for semilinear second-order neutral evolution equations with infinite delay

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.

Full discretization error analysis of exponential integrators for semilinear wave equations

In this article we prove full discretization error bounds for semilinear second-order evolution equations. We consider exponential integrators in time applied to an abstract nonconforming semidiscretization in space. Since the fully discrete schemes involve the spatially discretized semigroup, a crucial point in the error analysis is to eliminate the continuous semigroup in the representation of the exact solution. Hence, we derive a modified variation-of-constants formula driven by the spatially discretized semigroup which holds up to a discretization error. Our main results provide bounds for the full discretization errors for exponential Adams and explicit exponential Runge–Kutta methods. We show convergence with the stiff order of the corresponding exponential integrator in time, and errors stemming from the spatial discretization. As an application of the abstract theory, we consider an acoustic wave equation with kinetic boundary conditions, for which we also present some numerical experiments to illustrate our results.

Riemannian Formulation of Pontryagin’s Maximum Principle for the Optimal Control of Robotic Manipulators

In this work, we consider robotic systems for which the mass tensor is identified to be the metric in a Riemannian manifold. Cost functional invariance is achieved by constructing it with the identified metric. Optimal control evolution is revealed in the form of a covariant second-order ordinary differential equation featuring the Riemann curvature tensor that constrains the control variable. In Pontryagin’s framework of the maximum principle, the cost functional has a direct impact on the system Hamiltonian. It is regarded as the performance index, and optimal control variables are affected by this fundamental choice. In the present context of cost functional invariance, we show that the adjoint variables are the first-order representation of the second-order control variable evolution equation. It is also shown that adding supplementary invariant terms to the cost functional does not modify the basic structure of the optimal control covariant evolution equation. Numerical trials show that the proposed invariant cost functionals, as compared to their non-invariant versions, lead to lower joint power consumption and narrower joint angular amplitudes during motion. With our formulation, the differential equations solver gains stability and operates dramatically faster when compared to examples where cost functional invariance is not considered.

A Galerkin approach to solitary wave propagation for the second-order nonlinear evolution equation based on quartic B-spline functions

This article aims to produce efficient numerical results using the Galerkin approach and quartic B-spline functions for the one-dimensional second-order nonlinear evolution equation. The quartic B-spline functions are used as the shape and weight functions for the Galerkin method. The time derivative and space parameters are discretized by the finite difference and Crank–Nicolson schemes, respectively. A Galerkin approach is investigated by propagating waves at various times and comparisons are made with published papers. The obtained results show that the scheme presented good results and conservations laws are all adequately obeyed.

A Qualitative Study on Second-Order Nonlinear Fractional Differential Evolution Equations with Generalized ABC Operator

This research paper is dedicated to an investigation of an evolution problem under a new operator (g-Atangana–Baleanu–Caputo type fractional derivative)(for short, g-ABC). For the proposed problem, we construct sufficient conditions for some properties of the solution like existence, uniqueness and stability analysis. Existence and uniqueness results are proved based on some fixed point theorems such that Banach and Krasnoselskii. Furthermore, through mathematical analysis techniques, we analyze different types of stability results. The symmetric properties aid in identifying the best strategy for getting the correct solution of fractional differential equations. An illustrative example is discussed for the control problem.

Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

In a Hilbert space mathcal{H}, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A= nabla f +B , where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as tto +infty of the generated trajectories towards the zeros of nabla f +B. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

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.