Asymptotic Behavior Of Solutions Research Articles

On the asymptotic behavior of solutions to bilinear Caputo stochastic fractional differential equations

In this paper, we focus on investigating the asymptotic behavior of solutions in a mean square sense to bilinear Caputo stochastic fractional differential equations (CSFDEs). The main tools in the proof include a variation of the constant formula for CSFDEs, the Jordan normal form of a matrix, the summation formula of Djrbashian type, and constructing a weighted norm in the associated Banach space.

Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation on the Semi-Axis with a Potential Which Slowly Depends on Time

Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation on the Semi-Axis with a Potential Which Slowly Depends on Time

Asymptotic smoothness effects and global attractor for a peridynamic model with energy damping

AbstractIn this paper, we consider a peridynamic model with energy damping inspired by the works of Balakrishnan and Taylor on “damping models” based on the instantaneous total energy of the system. We study the asymptotic behavior of solutions, in the sense of attractors, of these peridynamic models in suitable phase space; more precisely, we prove a result of existence and characterization of compact global attractors with a nonlinear strongly continuous semigroup approach based in the asymptotic smoothness property thanks to Chueshov and Lasiecka and Nakao's lemma.

Asymptotically autonomous robustness of random attractors for the stochastic wave lattice equations with random viscosity

In this paper, we focus on the asymptotic behavior of solutions to the non-autonomous stochastic wave lattice systems with random viscosity and multiplicative white noise. Under proper conditions, we first prove the existence and uniqueness of the pullback random attractors and then show their backward compactness in a weighted space. Finally, we establish the asymptotically autonomous robustness of pullback random attractors when the nonlinearity is bounded and the time-dependent forcing term converges to the time-independent counterpart.

Solutions to the nonlinear obstacle problem with compact contact sets

For the obstacle problem with a nonlinear operator, we characterize the space of global solutions with compact contact sets. This is achieved by constructing a bijection onto a class of quadratic polynomials describing the asymptotic behavior of solutions.

Long‐time asymptotics of solution for the fifth‐order modified KdV equation in the presence of discrete spectrum

AbstractWe investigate the Cauchy problem of an integrable focusing fifth‐order modified Korteweg–de Vries (KdV) equation, which contains the fifth‐order dispersion and relevant higher order nonlinear terms. The long‐time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth‐order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Asymptotic behavior for Kirchhoff type stochastic plate equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of Kirchhoff type stochastic nonlinear plate equations with dispersive and viscosity dissipative terms driven additive noise defined on the entire space Rn. The existence, uniqueness as well as upper semi-continuity of pullback random attractors are proved in H2(Rn)×H1(Rn).

Existence and asymptotic behavior of solutions for quasilinear Schrödinger equations involving p-Laplacian

Existence and asymptotic behavior of solutions for quasilinear Schrödinger equations involving p-Laplacian

Non‐uniformly exponential stability for time‐varying continuous‐time linear system

Exponential stability analysis is one of the most important problems in the study of the asymptotic behavior of solutions of continuous‐time linear systems. In comparison with the classical exponential stability, weaker criteria of (strongly) non‐uniformly exponential stability for the time‐varying linear systems are derived in this paper. By a nonsingular linear state transformation, a (strongly) non‐uniformly exponentially stable system can be equivalent to a simple one with the coefficient matrix whose measure is bounded by negative constants.

Asymptotic behavior of solutions of biharmonic Stefan’s problem

Asymptotic behavior of solutions of biharmonic Stefan’s problem

Existence and Asymptotic Behavior of Solutions for a Quasilinear Schrödinger–Poisson System in ℝ3 with a General Nonlinearity

Existence and Asymptotic Behavior of Solutions for a Quasilinear Schrödinger–Poisson System in ℝ3 with a General Nonlinearity

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

A constructive approach for investigating the stability of incommensurate fractional differential systems

This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.

Kamenev-Type Criteria for Testing the Asymptotic Behavior of Solutions of Third-Order Quasi-Linear Neutral Differential Equations

This paper aims to study the asymptotic properties of nonoscillatory solutions (eventually positive or negative) of a class of third-order canonical neutral differential equations. We use Riccati substitution to reduce the order of the considered equation, and then we use the Philos function class to obtain new criteria of the Kamenev type, which guarantees that all nonoscillatory solutions converge to zero. This approach is characterized by the possibility of applying its conditions to a wider area of equations. This is not the only aspect that distinguishes our results; we also use improved relationships between the solution and the corresponding function, which in turn is reflected in a direct improvement of the criteria. The findings in this article extend and generalize previous findings in the literature and also improve some of these findings.

Asymptotic Behavior of Solutions of a Complete Second-Order Integro-Differential Equation

Asymptotic Behavior of Solutions of a Complete Second-Order Integro-Differential Equation

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra–Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schläfli integral of the Volterra–Bessel function. In this respect, we study the asymptotic analysis of the Volterra–Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

Novel criteria for stability and convergence of solutions for non-autonomous nonlinear systems

In this paper, we give a new integral inequality which is used to study the asymptotic behaviour of solutions for a class of nonlinear dynamic systems with small perturbation using a numerical approach. We provide some new results on the stability of perturbed systems where necessary and sufficient condition is derived. We show that the perturbed nonlinear system can be globally uniformly practically asymptotically stable provided that the bound of perturbation is small enough. A numerical example is presented to illustrate the validity of the main result.