Decay Rates Of Solutions Research Articles

Classification of solutions of elliptic equations arising from a gravitational O(3) gauge field model

In this paper, we study an elliptic equation arising from the self-dual Maxwell gauged O(3) sigma model coupled with gravity. When the parameter τ equals 1 and there is only one singular source, we consider radially symmetric solutions. There appear three important constants: a positive parameter a representing a scaled gravitational constant, a nonnegative integer N1 representing the total string number, and a nonnegative integer N2 representing the total anti-string number. The values of the products aN1,aN2∈[0,∞) play a crucial role in classifying radial solutions. By using the decay rates of solutions at infinity, we provide a complete classification of solutions for all possible values of aN1 and aN2. This improves previously known results.

Exact rate of decay for solutions to damped second order ODE's with a degenerate potential

We prove exact rate of decay for solutions to a class of second order ordinary differential equations with degenerate potentials, in particular, for potential functions that grow as different powers in different directions in a neigborhood of zero. As a tool we derive some decay estimates for scalar second order equations with non-autonomous damping.

Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping

In this paper, we consider the large time behavior of the strong solutions to the three dimensional compressible viscoelastic flows with damping. Based on the energy method and spectral analysis, we analyze the influences of the damping on the global existence and decay rates of compressible viscoelastic flows under some small assumptions in H3-framework. Compared with the time decay rates of solutions to the compressible viscoelastic flows in [1], our results imply that the friction of the damping is stronger than the dissipation effect of the viscosities.

Remarks on L^{2} decay of solutions for the third-grade non-Newtonian fluid flows in mathbb{R}^{3}

This paper is concerned with the improved L^{2} decay for solutions of a class of the third-grade non-Newtonian fluid flows in mathbb {R}^{3}. By developing the classic Fourier splitting methods, we prove the non-uniform decay of solutions when u_{0}in L^{2}(mathbb {R}^{3}) and improve algebraic decay rates of solutions as (1+t)^{{-frac{3}{2}}({frac{1}{r}}-{frac{1}{2}})-frac{1}{2} } when the initial data satisfy some moment condition. The results extend the previous result by Zhao (Nonlinear Anal., Real World Appl. 15:229-238, 2014).

Well-posedness and energy decay for Timoshenko systems with discrete time delay under frictional damping and/or infinite memory in the displacement

In this paper, we consider a vibrating system of Timoshenko-type in a bounded one-dimensional domain with discrete time delay and complementary frictional damping and infinite memory controls all acting on the transversal displacement. We show that the system is well-posed in the sens of semigroup and that, under appropriate assumptions on the weights of the delay and the history data, the stability of the system holds in case of the equal-speed propagation as well as in the opposite case in spite of the presence of a discrete time delay, where the decay rate of solutions is given in terms of the smoothness of the initial data and the growth of the relaxation kernel at infinity. The results of this paper extend the ones obtained by the present author and Messaoudi in (Acta Math Sci 36:1–33, 2016) to the case of presence of discrete delay.

On the exponential stability of solutions of periodic systems of the neutral type with several delays

We obtain conditions for the exponential stability of the zero solution of linear periodic systems of differential equations of the neutral type with several constant delays, which are stated in terms of a Lyapunov–Krasovskii functional of a special form. We derive estimates that specify the decay rate of solutions at infinity.

Polynomial Decay of Solutions to the Cauchy Problem for a Petrovsky–Petrovsky System in R n $\mathbb{R}^{n}$

This paper describes a polynomial decay rate of solution for a coupled system of Petrovsky equations in Rn$\mathbb{R}^{n}$ with infinite memory acting in the first equation. The weighted spaces and results in Guesmia (Appl. Anal. 94(1):184---217, 2015) are also used. The main contributions here is to show that the infinite memory lets our problem still dissipative and that the system is not exponentially stable in spite of the kernel in the memory term is sub-exponential.

Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the Navier- Stokes equations.We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, me characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.

Estimates for Solutions of One Class of Nonlinear Neutral Type Systems with Several Delays

We consider a class of systems of nonlinear differential equations of neutral type with several delays. We obtain conditions of exponential stability of the zero solution and establish estimates characterizing the exponential decay rate of solutions at infinity. Bibliography: 14 titles.

Global existence and optimal decay rates of solutions for compressible Hall-MHD equations

In this paper, we are concerned with global existence and optimal decay rates ofsolutions for the compressible Hall-MHD equations in dimension three.First, we prove the global existence ofstrong solutions by the standard energy method under the conditionthat the initial data are close to the constant equilibrium state in $H^2$-framework.Second, optimal decay rates of strong solutions in $L^2$-norm are obtainedif the initial data belong to $L^1$ additionally.Finally, we apply Fourier splitting method bySchonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates forhigher order spatial derivatives of classical solutions in $H^3$-framework,which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].

Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.

Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem

Abstract We obtain upper bounds for the decay rate for solutions to the nonlocal problem ∂ t u ( x , t ) = ∫ R n J ( x , y ) | u ( y , t ) − u ( x , t ) | p − 2 ( u ( y , t ) − u ( x , t ) ) d y with an initial condition u 0 ∈ L 1 ( R n ) ∩ L ∞ ( R n ) and a fixed p > 2 . We assume that the kernel J is symmetric, bounded (and therefore there is no regularizing effect) but with polynomial tails, that is, we assume a lower bounds of the form J ( x , y ) ≥ c 1 | x − y | − ( n + 2 σ ) , for | x − y | > c 2 and J ( x , y ) ≥ c 1 , for | x − y | ≤ c 2 . We prove that ∥ u ( ⋅ , t ) ∥ L q ( R n ) ≤ C t − n ( p − 2 ) n + 2 σ ( 1 − 1 q ) for q ≥ 1 and t large. MSC: 35K05, 45P05, 35B40.

Boundary Feedback Stabilization of Kirchhoff-Type Timoshenko System

In this work, the stabilization problem of the nonlinear vibrating Timoshenko systems of Kirchhoff-type with boundary control conditions is considered. By virtue of the multiplier method, the explicit energy decay rates for solutions of the system are established, depending on boundary control feedback. In the view of control, the result of this work implies that, by choosing suited feedback boundary controls, the Kirchhoff-type Timoshenko system can be achieved by various decay rates, not only exponential and polynomial.

Positive solutions to integral systems with weight and Bessel potentials

In this paper, we consider the integral system with weight and the Bessel potentials:{u(x)=∫Rngα(x−y)u(y)pv(y)q|y|σdy,v(x)=∫Rngα(x−y)v(y)pu(y)q|y|σdy, where u,v>0, σ⩾0, 0<α<n, p+q=γ⩾2 and gα(x) is the Bessel potential of order α. First, we get the integrability by regularity lifting lemma. Then we also establish the regularity of the positive solutions. Afterwards, by the method of moving planes in integral forms, we show that the positive solutions are radially symmetric and monotone decreasing about the origin. Finally, by an extension of the idea of Lei [14] and analytical techniques, we get the decay rates of solutions when |x|→∞.

Improved decay rates for solutions for a multidimensional generalized Benjamin–Bona–Mahony equation

In this paper, we study the decay rates of solutions for the generalized Benjamin-Bona-Mahony equation in multi- dimensional space. For initial data in some L 1 -weighted spaces, we prove faster decay rates of the solutions. More precisely, using the Fourier transform and the energy method, we show the global existence and the convergence rates of the solutions under the smallness assumption on the initial data and we give better decay rates of the solutions. This result improves early works in J. Differential Equations 158(2) (1999), 314-340 and Nonlinear Anal. 75(7) (2012), 3385-3392.

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic-parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ L1-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.

Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains

The first initial-boundary value problem with the homogeneous Dirichlet boundary condition and a compactly supported initial function is considered for a model second-order anisotropic parabolic equation in a cylindrical domain D = (0,∞) × Ω. We find an upper bound that characterizes the dependence of the decay rate of solutions as t → ∞ on the geometry of the unbounded domain Ω ⊂ ℝn, n ≥ 3, and on nonlinearity exponents. We also obtain an estimate for the admissible decay rate of nonnegative solutions in unbounded domains; this estimate shows that the upper bound is sharp.

Decay rate of quasilinear wave equation with viscosity

This paper is concerned with the decay rate of solutions for a quasilinear wave equation with viscosity. We use a so-called energy perturbation method to establish decay rate of solutions in terms of energy norm for a class of nonlinear functions. With the help of a comparison lemma of differential inequalities, we establish a relationship between decay rate of solutions and f.

Global existence and asymptotic behavior of the solutions to the three-dimensional bipolar Euler–Poisson systems

In this paper, the global existence of smooth solutions for the three-dimensional bipolar hydrodynamic model is studied when the initial data are close to a constant state. The L 2 -decay rate of the solutions is also obtained. Our approach is based on the detailed analysis of the Green function of the linearized system and the elaborate energy estimates. As far as we known, it is the first result about the existence and L 2 -decay rate of global smooth solutions to the multi-dimensional bipolar hydrodynamic model.