Decay Rates Of Solutions Research Articles

Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects (E) $$ \left\{ \begin{aligned} & \psi _t = - (1 - \alpha )\psi - \theta _x + \alpha \psi _{xx} , \\ & \theta _t = - (1 - \alpha )\theta + \nu \psi _x + 2\psi \theta _x + \alpha \theta _{xx} , \\ \end{aligned} \right. $$ with initial data (I) $$ (\psi ,\theta )(x,0) = (\psi _0 (x),\theta _0 (x)) \to (\psi _ \pm ,\theta _ \pm )\quad {\text{as}}\quad x \to \pm \infty , $$ where α and ν are positive constants such that α < 1,ν < α (1−α). Through constructing a correct function % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK % aacaGGOaGaamiEaiaacYcacaWG0bGaaiykaaaa!3BB3! $$\hat \theta (x,t)$$ defined by (2.13) and using the energy method, we show % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci % GGZbGaaiyDaiaacchaaSqaaiaadIhacqGHiiIZtuuDJXwAK1uy0HMm % aeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab-1risbqabaGccaGGOaWaaq % qaaeaacaGGOaGaeqiYdKNaaiilaiabeI7aXjaacMcacaGGOaGaamiE % aiaacYcacaWG0bGaaiykamaaeeaabaGaey4kaScacaGLhWoadaabba % qaaiaacIcacqaHipqEdaWgaaWcbaGaamiEaaqabaaakiaawEa7aiaa % cYcacqaH4oqCdaWgaaWcbaGaamiEaaqabaGccaGGPaGaaiikaiaadI % hacaGGSaGaamiDaiaacMcadaabbaqaaiaacMcacqGHsgIRcaaIWaaa % caGLhWoaaiaawEa7aaaa!6701! $$\mathop {\sup }\limits_{x \in \mathbb{R}} (\left| {(\psi ,\theta )(x,t)\left| + \right.\left| {(\psi _x } \right.,\theta _x )(x,t)\left| {) \to 0} \right.} \right.$$ as % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk % ziUkabg6HiLcaa!3A45! $$t \to \infty $$ and the solutions decay with exponential rates. The same problem is studied by Tang and Zhao [10] for the case of (ψ±, θ±) = (0,0).

On Stability of Solutions to Quasilinear Periodic Systems of Differential Equations

We consider a quasilinear system of differential equations with periodic coefficients in the linear terms. We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity. The results are stated in terms of the integrals of the norm of a periodic solution to the Lyapunov differential equation.

Decay rate of solutions of a wave equation with damping and external force

Decay rate of solutions of a wave equation with damping and external force

Decay rate of solutions to hyperbolic system of first order

In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case. We obtain several decay estimates of solutions of a hyperbolic system of first order by different norms of initial data. In particular, the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions, if the initial data belongs to the assigned weighted Sobolev space. In the proof of the theorem we reduce the estimate of solutions of a hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order, and then establish the required estimate by using microlocal analysis.

Asymptotic stability and decay rate of solutions for a nonlinear wave equation with variable coefficients

This paper is concerned with investigating the global asymptotic behavior of the solution to a nonlinear wave equation with variable coefficients. Moreover an estimate of the rate of decay of the solution is obtained.

Critical Exponents for the Decay Rate of Solutions in a Semilinear Parabolic Equation

This paper is concerned with the Cauchy problem \(\) A solution u is said to decay fast if \(t ^{1/(p-1)} u \rightarrow 0\) as \(t \rightarrow \infty\) uniformly in R, and is said to decay slowly otherwise. For each nonnegative integer k, let \(\Lambda _k\) be the set of uniformly bounded functions on R which change sign k times, and let \(p_k>1\) be defined by \( p_k=1+2/(k+1)\). It is shown that any nontrivial bounded solution with \(u_0\in\Lambda _k\) decays slowly if \(1 < p < p_k\), whereas there exists a nontrivial fast decaying solution with \(u_0 \in\Lambda_k\) if \(p\gep_k\).

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials.

The rate of decay for solutions in nonlinear dissipative partial differential equations

We show, applying the method used by Dyer and Edmunds [Proc. London Math. Soc. 18 (1968) 169]. for the Navier-Stokes equations, that the rate of decay for certain dissipative partial differential equations can be bounded below by an exponential. In particular we show that this holds true for a generalised diffusion equation closely linked to the complex Ginzburg-Landau equation [M.V. Bartuccelli et al., Phys. Scr, in press] and that a similar method can be applied to the Kuramoto-Sivashinsky equation and the Burgers equation. We also discuss the implications of these results for the estimates of the dissipative length scale derived for dissipative partial differential equations using ladder methods.

Asymptotic stability and decay rate of solutions for a nonlinear fourth order wave equation

This paper is concerned with investigating the global asymptotic behavior of the zero solution of the initial-boundary value problem for a nonlinear fourth order wave equation. Moreover an estimate of the rate of decay of the solutions is obtained.

Saint-Venant decay rates for an isotropic inhomogeneous linearly elastic solid in anti-plane shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials.

Some a Priori Estimates for a Singular Evolution Equation Arising in Thin-Film Dynamics

This paper considers the singular evolution equation $u_t - \Delta \ln u = 0$, particularly the corresponding Cauchy problem. This equation arises in the study of thin film dynamics and as the formal limit as $m \to 0$ of the porous-medium equation. Through the use of local sup-estimates, similar to those for the porous-medium equation (see E. DiBenedetto and Y. L. Kwong, Intrinsic Harnack estimates and extinction profile for certain parabolic equations, Trans. Amer. Math. Soc., 330 (1992), pp. 783–811), and a global Harnack-type inequality, a critical decay rate for solutions in three or more space dimensions as $|x| \to \infty $ is found. In particular, if the initial data decays faster than this critical rate, there is no solution to the Cauchy problem.

On the decay rate of solutions of linear parabolic equations for infinitely increasing time

The first boundary value problem with zero boundary values is considered for a one-dimensional linear parabolic equation. If the equation is sufficiently close to the heat equation, the rate of decreasing for the solution is connected with the number of zero level lines of the solution, nonvanishing for all values of time. Bibliography: 4 titles.

On some nonlinear hyperbolic systems with damping boundary conditions

IN RECENT years the effect of boundary damping on the solutions to interior initial boundary value problems for hyperbolic equations has been extensively studied. In [ 111, Quinn and Russell considered the wave equation with damping boundary condition on a part of the boundary. It was proved by control-theoretic method that if the boundary where the boundary condition was posed was star-shaped, then the solution of the wave equation had polynomial decay rate. Later on, Chen in [l] got the exponential decay of solutions to the wave equation under the more restrictive assumptions on the boundary. And also, Lagnese in [6] got the exponential decay of solutions to the three-dimensional elastic wave equations with damping boundary condition on a part of the boundary provided the geometry of the boundary is suitably restricted. As for the nonlinear problem, Greenberg and Li in 131 considered the nonlinear conservation system in one space dimension with damping boundary condition and proved the global existence of smooth solution when the initial data are small. Shen and Zheng [14] and Nagasawa [9] proved the global existence of classical solutions to the system of one-dimensional thermoelasticity and to the system of one-dimensional nonlinear problem in higher space dimension, respectively. For the nonlinear problem in higher space dimension, to our knowledge, the works were done by Qin [lo] and Zajaczkowski [26] only for the nonlinear wave equation with damping boundary conditions based on Chen’s results in [l]. In this paper our aim is to derive the decay rate of solutions to the linear hyperbolic system of second order with damping boundary condition without any assumptions on the geometry of the boundary by using the spectral analysis method (cf. Section 2 below), and then to prove the global existence of smooth solutions to the corresponding nonlinear system with applications to nonlinear elastodynamics and to nonlinear acoustic wave equations. Our approach to obtaining the rate of decay of solutions to linear problem is completely different from [l], [6] and [lo].

Invariant manifolds with asymptotic phase

THIS PAPER is concerned with invariant manifolds allowing a generic approach to solutions (of the underlying autonomous differential system) only. This means. that any solution approaching such an invariant manifold M ultimately behaves like a particular solution on J4, in the sense that the difference n-(t) i(l) of the two solutions tends to 0 as t + CQ. In other words, only solutions on the stable manifold of M can approach M as f+ cc. Invariant manifolds with that property are said to have an asymptotic phase. The simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose Floquet multipliers lie all but one in the open unit disc of the complex plane. Malkin [12] proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold. More generally, a k-dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k Floquet multipliers have modulus less than 1 (see Hale & Stokes [9]). Conditions for a torus to be asymptotically stable with asymptotic phase are given by Samoilenko [13] and Coppel[5]. Invariant manifolds without any specified structure are considered by Fenichel [6,7] and in Kirchgraber’s note [ 111. Moreover, each center manifold has an asymptotic phase if the corresponding equilibrium is stable (see Carr & Al-Amood [4] and Henry [lo]). In any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold. In each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability. A negative answer to this question is given by the author in [l, 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase. In [l] the manifold is supposed to be compact while the noncompact case is treated in [2] under the restriction that the generating family of periodic solutions has an amplitude independent period. Recently Hale & Massatt [8] extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations. In this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result: A compact invariant manifold has an asymptotic phase if the manifold is hyperbolic (assumption Hl) and carries a flow parallel in the sense of our assumption H2. This includes most of the aforementioned results.

Rate of decay for solutions of stochastic differential equations

A theorem is given which demonstrates that solutions of a stochastic differential equation decay to zero at least as fast as a function ψ( t) provided L[ ln V] ⩽ (d dt) ( ln ψ) , where V is a Liapunov-like function and L is the operator associated with the stochastic equation. The result is an extension of a recent result of A. Friedman and M. Pinsky. A further generalization is discussed.

A Minimal Decay Rate for Solutions of Stable nTH Order Homogeneous Differential Equations with Constant Coefficients

A Minimal Decay Rate for Solutions of Stable nTH Order Homogeneous Differential Equations with Constant Coefficients