Decay property for second order hyperbolic systems of viscoelastic materials

We study a class of second order hyperbolic systems with dissipation which describes viscoelastic materials. The considered dissipation is given by the sum of the memory term and the damping term. When the dissipation is effective over the whole system, we show that the solution decays in L² at the rate t−n/4 as t→∞, provided that the corresponding initial data are in L²∩L¹, where n is the space dimension. The proof is based on the energy method in the Fourier space. Also, we discuss similar systems with weaker dissipation by introducing the operator (1−Δ)^−θ/2 with θ>0 in front of the dissipation terms and observe that the decay structure of these systems is of the regularity-loss type.     

A Liapunov functional for a singular integral equation

We consider a scalar integral equation where a∈L²[0,∞), while C(t,s) has a significant singularity, but is convex when t−s>0. We construct a Liapunov functional and show that g(t,x(t))−a(t)∈L²[0,∞) and that x(t)−a(t)→0 pointwise as t→∞. Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals.     

Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

On the wave equation with a large rough potential

We prove an optimal dispersive L^∞ decay estimate for a three-dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI

Painlevé's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2×2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ? of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P(t,∞):L²(0,∞)→L²(t,∞) be the orthogonal projection; then the Fredholm determinant τ(t)=det(I−P(t,∞)Γ?) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand-Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L²(0,∞); so det(I−Γ?P(t,∞)) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ?=1.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Remark on theL2 decay for weak solution to equations of non-newtonian incompressible fluids in the whole space (I)

We study the asymptotic behaviour of the certain class of non-Newtonian incompressible fluids-power law fluids in the whole space when the external force is zero. Assuming that initial data belong toL ¹∩L ² we prove thatL ² decay in time ist ^−1/4.

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Sunto Noi studiamo il comportamento asintotico in tutto lo spazio di una classe di fluidi incomprimibili non-Newtoniani con una legge ?fluids-power? in assenza di forze esterne. Assumendo che i dati iniziali appartengano aL ¹∩L ² noi proviamo che il decadimento nel tempo inL ² èt ^−1/4.

     

Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov–Bohm magnetic field

We consider two-dimensional Schrödinger operators H   with an Aharonov–Bohm magnetic field and an additional electric potential. We obtain an explicit leading term of the asymptotic expansion of the unitary group e^−itH$e^{-itH}$ for t→∞$t\to \infty$ in weighted L²$L^2$-spaces. In particular, we show that the magnetic field improves the decay of e^−itH$e^{-itH}$ with respect to the unitary group of non-magnetic Schrödinger operators, and that the decay rate in time is determined by the magnetic flux.     

Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U ₀?∈?H ^s (?)?∩?L ¹(?). In addition, by restricting the initial data to U ₀?∈?H ^s (?)?∩?L ^1,γ(?) and γ?∈?[0,?1], we can derive faster decay estimates with the decay rate improvement by a factor of t ^?γ/2.     

Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.     

Positive definite distributions and normed spaces

We answer a question of Alex Koldobsky. We show that for each −∞<p<2 and each n?3−p there is a normed space X of dimension n which embeds in Ls if and only if −n<s?p.     

Large time behavior of disturbed planar fronts in the Allen-Cahn equation

We consider the Allen-Cahn equation in Rn (with n?2) and study how a planar front behaves when arbitrarily large (but bounded) perturbation is given near the front region. We first show that the behavior of the disturbed front can be approximated by that of the mean curvature flow with a drift term for all large time up to t=+∞. Using this observation, we then show that the planar front is asymptotically stable in L^∞(Rn) under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. As a by-product of our analysis, we present a result of a rather general nature, which states that, for a large class of evolution equations, the unique ergodicity of the initial data is inherited by the solution at any later time.     

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved L ² decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.     

Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems

This paper is concerned with nonlinear stability of traveling wave fronts for a delayed reaction diffusion system. We prove that the traveling wave front is exponentially stable to perturbation in some exponentially weighted L^∞ spaces, when the difference between initial data and traveling wave front decays exponentially as x→−∞, but the initial data can be suitable large in other locations. Moreover, the time decay rates are obtained by weighted energy estimates.     

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p−L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature hypothesis.     

Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L²-norm. The numerical examples are given to illustrate the theoretical results.     

Stability of small‐amplitude shock profiles of symmetric hyperbolic‐parabolic systems

Combining pointwise Green's function bounds obtained in a companion paper [36] with earlier, spectral stability results obtained in [16], we establish nonlinear orbital stability of small‐amplitude Lax‐type viscous shock profiles for the class of dissipative symmetric hyperbolic‐parabolic systems identified by Kawashima [20], notably including compressible Navier‐Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in L^p with respect to small L¹ ∩ H³ perturbations, 2 ≤ p ≤ ∞. Our analysis extends and somewhat refines the approach introduced in [35] to treat stability of relaxation profiles. © 2004 Wiley Periodicals, Inc.