Asymptotic Behavior Of Solutions Research Articles

Long-time behavior of solutions for time-periodic reaction-diffusion equations and applications

This article concerns the asymptotic behavior of solutions for time-periodic reaction-diffusion equations with a drift term in one dimensional space. Assuming that drift term is decaying, we analyze the effect of the decaying rate of the drift term on the propagation speed of solutions. Also we show some applications of our results in high-dimensional domains. For more information see https://ejde.math.txstate.edu/Volumes/2024/69/abstr.html

Results on a nonlinear wave equation with acoustic and fractional boundary conditions coupling by logarithmic source and delay terms: Global existence and Asymptotic behavior of solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is notable for its capacity to model complex systems, contribute to the advancement of mathematical theory, and exhibit wide-ranging applicability to real-world problems. This paper investigates the global existence and general decay of solutions to a wave equation characterized by the inclusion of logarithmic source and delay terms, governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under various hypotheses, and the general decay behavior is established through the construction and application of a suitable Lyapunov function

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

Uniform boundedness and asymptotic behavior of solutions in a chemotaxis model for alopecia areata

Uniform boundedness and asymptotic behavior of solutions in a chemotaxis model for alopecia areata

Asymptotic behavior of solutions of nonlinear discrete systems in critical cases

AbstractThe paper is dedicated to the analysis of the asymptotic behavior of essentially nonlinear discrete systems whose linearization possesses eigenvalues on the unit circle. For these systems, the paper establishes sufficient conditions for the asymptotic stability regardless of terms higher than the third order. Moreover, it derives an asymptotic stability criterium for a reduced critical subsystem. The proof incorporates a constructive approach for deriving Lyapunov functions using the center manifold reduction and normal form method. The key result of the paper asserts that, given the obtained stability conditions, the solutions of the system converge to the origin with a polynomial decay rate. The paper also introduces a method for evaluating the coefficients of the decay rate estimate. To illustrate these findings, we apply the obtained result to analyzing nonlinear business cycle and predator-pray models.

Functions of a Complex Variable with a Large Parameter and Construction of Regions

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains comes down to the study of integrals of exponential functions containing a large parameter. Such integrals differ significantly from the integrals to which the saddle point method is applicable. The saddle point method is not applicable to study such integrals. Thus, the problem arises of constructing domains and choosing integration paths for studying such integrals. In this work, specific examples of integrals show the construction of regions in the complex plane and the choice of integration paths. The chosen integration paths ensure that the integrals are bounded over a large parameter as this parameter tends to infinity. When constructing the domain and choosing integration paths, level lines of some harmonic functions that have zeros and singular points were used. The principle of symmetry is also used. In early works, cases were considered when the eigenvalues ​​of the first approximation matrix of a singularly perturbed equation had only zeros or only poles. Cases where the eigenvalues ​​have zeros and poles were not considered.

On the Oscillatory Behavior of Solutions of Second-Order Damped Differential Equations with Several Sub-Linear Neutral Terms

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results.

The asymptotic behavior of constant sign and nodal solutions of [formula omitted]-Laplacian problems as [formula omitted] goes to 1

In this paper we study the asymptotic behavior of solutions to the (p,q)-equation −Δpu−Δqu=f(x,u)inΩ,u=0on∂Ω,as p→1+, where N≥2, 1<p<q<1∗≔N/(N−1) and f is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1,q)-Laplace problem given by −div∇u|∇u|−Δqu=f(x,u)inΩ,u=0on∂Ω,has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space W01,q(Ω) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(Ω) of all functions of bounded variation. As far as we know this is the first work dealing with (1,q)-Laplace problems even in the direction of constant sign solutions.

The existences and asymptotic behavior of solutions to stochastic semilinear generalized Rayleigh–Stokes equation with delays

We consider a functional stochastic delay semilinear Rayleigh–Stokes equation involving Riemann–Liouville derivative. Our aim is using the resolvent theory, fixed point argument to prove the global solvability and gives some sufficient conditions to ensure the asymptotic stability of mild solutions in the mean square moment.

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

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.

Results from a Nonlinear Wave Equation with Acoustic and Fractional Boundary Conditions Coupling by Logarithmic Source and Delay Terms: Global Existence and Asymptotic Behavior of Solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

Asymptotic behavior of solutions of the nonlinear Beltrami equation with the Jacobian

Asymptotic behavior of solutions of the nonlinear Beltrami equation with the Jacobian

Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients

We consider the following Neumann boundary problem with a nonlocal Kirchhoff term where is a bounded smooth domain and and are sign‐changing coefficients which play important impact on the study. By using variational method and dividing corresponding Nehari manifold into two disjoint subsets, we study the existence of two positive solutions. To show the multiplicity of positive solutions, we find a set whose elements can be mapped to two subsets of Nehari manifold, which is also the key to proving strong convergence for minimizing sequences. Finally, the asymptotic behavior of solutions is studied as parameter tends to 0. Different from Dirichlet boundary problem, we show that the two solutions of Neumann problem tend to two different positive constants, respectively.

Asymptotic behavior of solutions to a fourth-order degenerate parabolic equation

Asymptotic behavior of solutions to a fourth-order degenerate parabolic equation

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

Global solvability and asymptotic behavior of solutions for a fully parabolic nutrient taxis system

In this paper, we consider the fully parabolic nu’trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t &amp;gt; 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights.

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.