Decay Rates Of Solutions Research Articles

Analysis of Nonlinear Schrödinger Equations with Time-Dependent Coefficients: Stability and Decay Properties

This study investigates the stability and decay properties of solutions to nonlinear Schrödinger equations (NLSEs) with time-dependent coefficients. Employing a blend of analytical and numerical methods, we delve into how temporal variations in coefficients influence the dynamics of wave functions. Our analysis reveals that time-dependent coefficients significantly affect the stability and decay rates of solutions, uncovering conditions that lead to either enhanced stability or accelerated decay. The findings highlight the critical role of coefficient temporality in dictating the behavior of NLSE solutions. These insights not only advance our theoretical understanding of NLSEs but also bear implications for practical applications in fields modeled by these equations. Our research opens avenues for exploiting time-dependent behaviors in designing systems with desired dynamical properties.

Stability and rate of decay for solutions to stochastic differential equations with Markov switching

In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results. For more information see https://ejde.math.txstate.edu/Volumes/2024/01/abstr.html

The influence of damping on the asymptotic behavior of solution for laminated beam

<p>This paper dealt with a laminated beam system along with structural damping, past history, distributed delay, and in the presence of both temperatures and micro-temperatures effects. The damping terms left the system dissipative. Employing the semigroup approach, we established the existence and uniqueness of the solution. Additionally, with the help of convenient assumptions on the kernel, we demonstrated a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagation. The main aim was to address how specific behaviors of the system were related to memory and delays. We aimed to investigate the joint impact of an infinite memory, distributed delay and micro-temperature effects on the system. We found a new relationship between the decay rate of solution and the growth of g at infinity. The objective was to find studies that use no- trivial results and their applications to relevant problems from mathematical physics.</p>

Optimal decay rate of the incompressible Navier–Stokes–Maxwell system with Ohm’s law

This work is to establish optimal time decay rate of global classical solutions to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law and the incompressible Navier–Stokes–Maxwell system with Ohm’s law in R3. The results show that the optimal time decay rate for the velocity of these two systems achieves the same as that of the incompressible Navier–Stokes equation. In other words, Ohm’s law can offset the negative effects brought by the Maxwell equation on the decay rate of solutions. This is interesting compared to the compressible Navier–Stokes system, whose time decay rate decreases when it is coupled with the self-consistent Maxwell system.

Optimal decay rates of higher–order derivatives of solutions for the compressible nematic liquid crystal flows in $ \mathbb R^3 $

<abstract><p>In this paper, we are concerned with optimal decay rates of higher–order derivatives of the smooth solutions to the $ 3D $ compressible nematic liquid crystal flows. The main novelty of this paper is three–fold: First, under the assumptions that the initial perturbation is small in $ H^N $–norm $ (N\geq3) $ and bounded in $ L^1 $–norm, we show that the highest–order spatial derivatives of density and velocity converge to zero at the $ L^2 $–rates is $ (1+t)^{-\frac{3}{4}-\frac{N }{2 }} $, which are the same as ones of the heat equation, and particularly faster than the $ L^2 $–rate $ (1+t)^{-\frac{1}{4}-\frac{N }{2 }} $ in [J.C. Gao, et al., J. Differential Equations, 261: 2334-2383, 2016]. Second, if the initial data satisfies some additional low frequency assumption, we also establish the lower optimal decay rates of solution as well as its all–order spatial derivatives. Therefore, our decay rates are optimal in this sense. Third, we prove that the lower bound of the time derivatives of density, velocity and macroscopic average converge to zero at the $ L^2 $–rate is $ (1+t)^{-\frac{5}{4}} $. Our method is based on low-frequency and high-frequency decomposition and energy methods.</p></abstract>

Optimal decay rates of solutions to hyperbolic conservation laws with damping

Optimal decay rates of solutions to hyperbolic conservation laws with damping

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

Energy Decay Estimates of Solutions for Viscoelastic Damped Wave Equations in $$\mathbf {R}^{n}$$

In the present work, we investigate the global existence and uniform decay rates of solutions to the Cauchy problem in $$\mathbf {R}^{n}$$ ( $$n\ge 1)$$ related to the dynamic behavior of evolution equations accounting for rotational inertial forces along with a linear viscoelastic memory damping arising in viscoelastic materials. Under certain conditions on the initial data and on the kernel, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates in higher Sobolev space of the solution are provided. The proof is carried out by means of the point-wise decay estimates of the solution in the Fourier space.

On the existence of unique global-in-time solutions and temporal decay rates of solutions to some non-Newtonian incompressible fluids

In this paper, we deal with a class of incompressible non-Newtonian fluids. We first give some conditions to the viscous part of the stress tensor to set our model. We then show that there exists a unique regular solution globally in time if $$u_{0}\in L^{2}\cap \dot{B}^{1}_{\infty ,1}$$ and is sufficiently small in $$\dot{B}^{1}_{\infty ,1}$$ . We finally derive temporal decay rates of the solution which are consistent with the decay rates of the linear part of our model.

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

<p style='text-indent:20px;'>The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.</p>

Well-posedness and stability for Bresse-Timoshenko type systems with thermodiffusion effects and nonlinear damping

<abstract><p>Nonlinear Bresse-Timoshenko beam model with thermal, mass diffusion and theormoelastic effects is studied. We state and prove the well-posedness of problem. The global existence and uniqueness of solution is proved by using the classical Faedo-Galerkin approximations along with two a priori estimates. We prove an exponential stability estimate under assumption $ (2.3)_{1} $ and polynomial decay rate for solution under $ (2.3)_{2} $, by using a multiplier technique combined with an appropriate Lyapuniv functions.</p></abstract>

The decay rates of solutions to a chemotaxis-shallow water system

The decay rates of solutions to a chemotaxis-shallow water system

Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-Poisson system with unequal viscosities. Under the assumption that the H3 norm of the initial data is small but its higher order derivatives can be arbitrarily large, the global existence and uniqueness of smooth solutions are obtained by an ingenious energy method. Moreover, if additionally, the $${\dot H^{- s}}\left({{1 \over 2} \leqslant s < {3 \over 2}} \right)\,\,{\rm{or}}\,\,\dot B_{2,\infty }^{- s}\left({{1 \over 2} < s \leqslant {3 \over 2}} \right)$$ norm of the initial data is small, the optimal decay rates of solutions are also established by a regularity interpolation trick and delicate energy methods.

General Decay Rate of Solution for Love-Equation with Past History and Absorption

In the present paper, we consider an important problem from the point of view of application in sciences and engineering, namely, a new class of nonlinear Love-equation with infinite memory in the presence of source term that takes general nonlinearity form. New minimal conditions on the relaxation function and the relationship between the weights of source term are used to show a very general decay rate for solution by certain properties of convex functions combined with some estimates. Investigations on the propagation of surface waves of Love-type have been made by many authors in different models and many attempts to solve Love’s equation have been performed, in view of its wide applicability. To our knowledge, there are no decay results for damped equations of Love waves or Love type waves. However, the existence of solution or blow up results, with different boundary conditions, have been extensively studied by many authors. Our interest in this paper arose in the first place in consequence of a query for a new decay rate, which is related to those for infinite memory ϖ in the third section. We found that the system energy decreased according to a very general rate that includes all previous results.

Decay estimate and global existence of a semilinear Mindlin-Timoshenko plate system with full frictional damping in the whole space

In this paper, we are interested in a semilinear Mindlin-Timoshenko system with an irrotationality condition for the angle variables. Under the smallness assumption on the initial data, the decay rate of solutions is obtained by using both the multiplier method in the Fourier space and the fundamental solution method. Moreover, we also prove the global existence of solutions by the standard method.

Global Existence and Asymptotic Stability of 3D Generalized Magnetohydrodynamic Equations

In this paper, we study the global existence and asymptotic dynamics of generalized magnetohydrodynamic equations in $${\mathbb {R}}^3$$, in which the dissipation terms are $$-\eta (-\Delta )^\alpha $$ and $$-\mu (-\Delta )^\beta $$, $$0<\alpha ,\,\beta <1$$. With the help of combining the local existence and the a priori estimates, we establish the global existence and uniqueness of solution with small initial data. Moreover, we obtain the asymptotic decay rates of solutions by the method of energy estimates.

Time decay rate of solutions to the tropical climate model equations in ℝ n

This paper is devoted to study the time decay of tropical climate model. We first establish the optimal decay rates in of the weak solutions with large initial data in and obtain the existence and uniqueness of global smooth solutions with small initial data. In addition, taking advantage of Fourier splitting method, we get the decay rates in for higher order derivatives of the smooth solutions with small initial data.

Estimates of the Exponential Decay of Solutions to Linear Systems of Neutral Type with Periodic Coefficients

We consider the class of linear systems of delay differential equations with periodic coefficients. Using a special class of Lyapunov–Krasovskiĭ functionals, we establish conditions for the exponential stability of the zero solution and obtain estimates characterizing the exponential decay rate of solutions at infinity.

Decay of solutions to anisotropic conservation laws with large initial data

In this paper, we study the large time behavior of solutions to the Cauchy problem for the anisotropic conservation laws in two dimensional space. Without any smallness assumption on the initial data, the decay rates of solutions in L2 space and homogeneous Sobolev space H˙γ are obtained by using the method of time-frequency decomposition and the classical energy method.

Decay of solutions to parabolic-type problem with distributed order Caputo derivative

We consider the decay of solution to a fractional diffusion equation with distributed order Caputo derivative. We assume that the elliptic operator is time-dependent and that the weight function, contained in the definition of the distributed order Caputo derivative, is just integrable. We establish the relation between behavior of the weight function near zero and the decay rate of solution.