Asymptotic Analysis - Volume 81, issue 2

Authors: Skujus, M.

Article Type: Research Article

Abstract: A time-periodic Stokes problem is studied in the domain with cylindrical outlets to infinity. A basis in the set of solutions to the homogeneous problem is constructed. This basis possesses several properties that are essential for the setting correct asymptotic conditions at infinity.

Keywords: cylindrical outlets to infinity, homogeneous time-periodic Stokes problem

DOI: 10.3233/ASY-2012-1118

Citation: Asymptotic Analysis, vol. 81, no. 2, pp. 93-119, 2013

Authors: Soloviev, A.A.

Article Type: Research Article

Abstract: It is shown that the dominant asymptotic term of solution to the problem ut +uux =K*u−u, u|t=0 =u0 , where K(x)=(1/2)e−|x| , is a function u0 (x,t)=2ζ(x,t)(1−∫−∞ x ζ(s,t) ds)−1 with ζ(x,t)=M0 (h)ex2 /4t /√4πt, M0 (h)=∫−∞ ∞ h(s) ds and h(x)=(1/2)u0 (x)exp {−(1/2)∫−∞ x u0 (s) ds}, and the function u0 (x,t) coincides with the principal asymptotic term of solution to the Burgers equation with initial function u0 .

Keywords: radiating gas, initial value problem, asymptotic behavior

DOI: 10.3233/ASY-2012-1122

Citation: Asymptotic Analysis, vol. 81, no. 2, pp. 121-134, 2013

Authors: Petrini, Milena

Article Type: Research Article

Abstract: We deal with an hyperbolic degenerate system with periodic oscillations in the lower order term and principal part with characteristics of variable multiplicity. In the homogenization limit the effective system does not present a memory term, however it loses the property of energy conservation. A spectral analysis on the Bloch space is worked out for the system in some particular case of oscillating term. Through a decomposition in Bloch waves we determine the transport for the limiting energy density related to the solution a Cauchy problem with oscillatory initial data.

Keywords: homogenization, spectral theory and eigenvalues problems for partial differential operators, Wigner measures

DOI: 10.3233/ASY-2012-1125

Citation: Asymptotic Analysis, vol. 81, no. 2, pp. 135-156, 2013

Authors: Vodev, Georgi

Article Type: Research Article

Abstract: We prove optimal resolvent estimates for self-adjoint operators of the form P(h)=(ih∇+b(x,h))2 +V(x,h) on L2 (Rn ), n≥3, where 0<h<<1 is a semi-classical parameter and the coefficients have very little regularity with respect to the radial variable only.

Keywords: resolvent, Schrödinger operators

DOI: 10.3233/ASY-2012-1134

Citation: Asymptotic Analysis, vol. 81, no. 2, pp. 157-170, 2013

Authors: Goldstein, Jerome A. | Reyes, Guillermo

Article Type: Research Article

Abstract: We prove an asymptotic energy equipartition result for abstract damped wave equations of the form utt +2F(S)ut +S2 u=0, where S is a strictly positive self-adjoint operator and the damping operator F(S) is “small”. This means that under certain assumptions, the ratio of suitably modified kinetic and potential energies, K˜ (t)/P˜ (t), tends to 1 as t→∞ for all nonzero solutions u(t) of the equation. Here, K˜ (t) and P˜ (t) are conveniently weighted versions of the usual kinetic and potential energies of the associated undamped equation. Previous results, concerning the undamped case and the scalar-damped one, are …particular cases. We propose an extension of the concepts of hyperbolicity and unitarity that allows one to consider the equipartition property in a more general setting. Some examples involving PDEs, as well as pseudo-differential equations, are given. Show more

Keywords: abstract wave equations, energy equipartition

DOI: 10.3233/ASY-2012-1124

Citation: Asymptotic Analysis, vol. 81, no. 2, pp. 171-187, 2013