Asymptotic Behavior Of Solutions Research Articles

Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

On asymptotic behavior of solutions of stochastic differential equations in multidimensional space

Consider the multidimensional SDE dX(t) = a(X(t)) dt + b(X(t)) dW(t). We study the asymptotic behavior of its solution X(t) as t → ∞, namely, we study sufficient conditions of transience of its solution X(t), stabilization of its multidimensional angle X(t)/|X(t)|, and asymptotic equivalence of solutions of the given SDE and the following ODE without noise: dx(t) = a(x(t)) dt.

Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime

Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime

Existence and asymptotic behavior of solutions for non-linear wave equations of Kirchhoff type with viscoelasticity

"In this paper we discuss the global existence and the asymptotic behavior of solutions of an initial boundary value problem of a non-linear wave equation of Kirchhoff type. "

Mass concentration for Ergodic Choquard Mean-Field Games

We study concentration phenomena in the vanishing viscosity limit for second-order stationary Mean-Field Games systems defined in the whole space RN with Riesz-type aggregating coupling and external confining potential. In this setting, every player of the game is attracted toward congested areas and the external potential discourages agents to be far away from the origin. Assuming some restrictions on the strength of the attractive nonlocal term depending on the growth of the Hamiltonian, we study the asymptotic behavior of solutions in the vanishing viscosity limit. First, we obtain existence of classical solutions to potential-free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.

A precise asymptotic description of half‐linear differential equations

AbstractWe study asymptotic behavior of solutions of nonoscillatory second‐order half‐linear differential equations. We give (in some sense optimal) conditions that guarantee generalized regular variation of all solutions, where no sign condition on the potential is assumed. For all of these solutions, we establish precise asymptotic formulas, where positive as well as negative potential is considered. We examine, as consequences, also equations with regularly varying coefficients, or with the coefficients viewed as perturbations of exponentials, or the equations under certain critical (double roots) settings. We make also asymptotic analysis of Poincaré–Perron solutions. Many of our results are new even in the linear case.

Motion planning and stabilization of nonholonomic systems using gradient flow approximations

Nonlinear control-affine systems described by ordinary differential equations with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems under the bracket-generating condition. This methodology is based on the approximation of a gradient-like dynamics by trajectories of the designed closed-loop system. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance.

Existence and asymptotic behavior of solutions for Kirchhoff equations with general Choquard-type nonlinearities

Existence and asymptotic behavior of solutions for Kirchhoff equations with general Choquard-type nonlinearities

Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations

We study the existence and multiplicity of solutions for the Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$ where \(\Omega\) is an open bounded and smooth domain in \(\mathbb R^{3}\), \(a>0 \) is the Bopp-Podolsky parameter. The unknowns are \(u,\phi:\Omega\to \mathbb R\) and \(\omega\in\mathbb R\). By using variational methods we show that for any \(a>0\) there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrodinger-Poisson system, as \(a\to 0\). For more information see https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html

Asymptotic behavior of solutions for a nonlinear viscoelastic higher-order p(x)-Laplacian equation with variable-exponent logarithmic source term

Asymptotic behavior of solutions for a nonlinear viscoelastic higher-order p(x)-Laplacian equation with variable-exponent logarithmic source term

Simultaneous uniqueness in determining the space‐dependent coefficient and source for a time‐fractional diffusion equation

This article concerns the uniqueness of an inverse problem of simultaneously identifying the space‐dependent coefficient and source in a one‐dimensional time‐fractional diffusion equation with derivative order and the zero Neumann boundary value. By additional boundary measurements, we first obtain the uniqueness of the coefficient from the Laplace transform and a transformation formula. Then, we further show the uniqueness of the source through the asymptotic behavior of solutions to the corresponding forward problem. The result shows that the uniqueness of the simultaneous identification can be obtained under the condition that the prior information only on one set of parameters in the model is given other than that of two sets.

Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type

Abstract The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ 2 u ( k ) ± k α u m ( k ) = 0 , {\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where k k is the independent variable with values k = k 0 , k 0 + 1 , … k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em} , u : { k 0 , k 0 + 1 , … } → R u:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k 0 {k}_{0} is a fixed integer, and Δ 2 u ( k ) {\Delta }^{2}u\left(k) is its second-order forward difference. New conditions with respect to parameters α ∈ R \alpha \in {\mathbb{R}} and m ∈ R m\in {\mathbb{R}} , m ≠ 1 m\ne 1 , are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation y ″ ( x ) ± x α y m ( x ) = 0 . {y}^{^{\prime\prime} }\left(x)\pm {x}^{\alpha }{y}^{m}\left(x)=0. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.

New generalized Halanay inequalities and relative applications to neural networks with variable delays

The asymptotic behavior of solutions for a new class of generalized Halanay inequalities is studied via the fixed point method. This research provides a new approach to the study of the stability of Halanay inequality. To make the application of fixed point method in stability research more flexible and feasible, we introduce corresponding functions to construct an operator according to different characteristics of coefficients. The results obtained in this paper are applied to the stability study of a neural network system, which has high value in application. Moreover, three examples and simulations are given to illustrate the results. The conclusions in this paper greatly improve and generalize the relative results in the current literature.

Scattering and Asymptotic Behavior of Solutions to the Vlasov–Poisson System in High Dimension

Scattering and Asymptotic Behavior of Solutions to the Vlasov–Poisson System in High Dimension

Critical mass capacity for two-dimensional Keller–Segel model with nonlocal reaction terms

This paper deals with the parabolic–elliptic Keller–Segel system on , involving a source term of logistic type defined in terms of the mass capacity M 0 and the total mass of the individuals. We exhibit that the qualitative behaviour of solutions is decided by the mass capacity M 0 and the initial mass m 0. For general solutions, the existence of a global weak solution is proved under the assumption that both M 0 and m 0 are less than 8π, whereas there exist solutions blowing up in finite time under the hypotheses of either with any integrable initial data or accompanied with large initial data. Moreover, gives rise to a compromise that solutions exist globally and blow up as time goes to infinity. For radially symmetric solutions, we introduce a strategy of relegating the lack of mass conservation via a transformation to the density and then obtain that there are stationary solutions given by with λ > 0. Subsequently we prove that if the initial data is strictly below for some λ > 0, then the solution vanishes in as . If the initial data is strictly above for some λ > 0, then the solution either blows up in finite time or has a mass concentration at the origin as time goes to infinity. Finally, our results are complemented by numerical simulations that demonstrate the asymptotic behaviour of solutions.

Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem

Consider the generalized discounted Hamilton-Jacobi equation \[ λ a ( x ) u + H ( x , D u ) = c ( H ) , \lambda a(x)u+H(x,Du)=c(H), \] where a ( x ) a(x) may vanish or change the signs. Two examples are given in this paper showing that the viscosity solutions of the above equation may not converge as λ \lambda tends to zero.

Inverse scattering problem for the matrix modified Korteweg–de Vries equation with finite density type initial data

The theory of inverse scattering is developed to study the initial-value problem for the matrix modified Korteweg–de Vries (mKdV) equation with the 2m×2m(m≥1) Lax pairs and finite density type initial data. In direct scattering problem, by introducing a suitable uniform transformation we establish the proper complex z-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry. Moreover, the asymptotic behavior of the Jost eigenfunctions and scattering matrix are analyzed needed in inverse problem. Then, the generalized Riemann–Hilbert problem of the matrix mKdV equation is established by using the analyticity of the modified eigenfunctions and scattering coefficients, from which the reconstruction formula of potential function with reflection-less case is derived successfully. Finally, some interesting phenomena in scalar and matrix cases are presented, as well as theoretical analysis of asymptotic behavior of solutions in a appendix.

Asymptotic Behavior of Solutions for Multi-stable Equation with Dirichlet Boundary Condition

Asymptotic Behavior of Solutions for Multi-stable Equation with Dirichlet Boundary Condition

Asymptotic Behavior of Solutions to a Nonlinear Swelling Soil System with Time Delay and Variable Exponents

In this research work, we investigate the asymptotic behavior of a nonlinear swelling (also called expansive) soil system with a time delay and nonlinear damping of variable exponents. We should note here that swelling soils contain clay minerals that absorb water, which may lead to increases in pressure. In architectural and civil engineering, swelling soils are considered sources of problems and harm. The presence of the delay is used to create more realistic models since many processes depend on past history, and the delays are frequently added by sensors, actuators, and field networks that travel through feedback loops. The appearance of variable exponents in the delay and damping terms in this system allows for a more flexible and accurate modeling of this physical phenomenon. This can lead to more realistic and precise descriptions of the behavior of fluids in different media. In fact, with the advancements of science and technology, many physical and engineering models require more sophisticated mathematical tools to study and understand. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools for studying such problems. By constructing a suitable Lyapunov functional, we establish exponential and polynomial decay results. We noticed that the energy decay of the system depends on the value of the variable exponent. These results improve on some existing results in the literature.