Asymptotic Analysis - Volume 16, issue 3-4

Authors: Zielinski, Lech

Article Type: Research Article

Abstract: We consider the Weyl formula describing the asymptotic behaviour of the number of eigenvalues N(\lambda) for elliptic boundary value problems. The remainder estimate of the form N(\lambda){\rm O}(\lambda^{-\mu}) is proved with \mu<r/(2m) in the case of operators of order 2m with Hölder coefficients of exponent r\in \,]0; 1] .

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 181-201, 1998

Authors: Kostin, I.N.

Article Type: Research Article

Abstract: The behaviour of trajectories of a nonlinear semigroup in the neighbourhood of a non‐hyperbolic stationary point is studied. The obtained results are used to derive an estimate for the rate of convergence of solutions of the Chafee–Infante problem to its attractor in the case of non‐hyperbolic zero stationary solution.

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 203-222, 1998

Authors: Donato, Patrizia | Gaudiello, Antonio | Sgambati, Luciana

Article Type: Research Article

Abstract: Consider the domain \varOmega_\varepsilon =\varOmega -T_\varepsilon obtained by removing a closed set T_\varepsilon of \varepsilon ‐periodic holes of size \varepsilon from a bounded open set \varOmega . We study the homogenization of the nonlinear problem \cases{-{\rm div}(A({x/\varepsilon})Du_\varepsilon)+ \gamma u_\varepsilon = H( {x/\varepsilon}, u_\varepsilon , Du_\varepsilon )& $\hbox{in }\varOmega_\varepsilon ,$\cr (A({x/\varepsilon})Du_\varepsilon)\cdot\underline\nu =0 &$\hbox{on } \Ncurpartial T_\varepsilon ,$\cr u_\varepsilon =0&$\hbox{on } \Ncurpartial \varOmega,$\cr u_\varepsilon\in H^1(\varOmega_\varepsilon )\cap L^\infty(\varOmega_\varepsilon ),&\cr} where H( y,s,\xi) is ]0,1[^n ‐periodic in y and has quadratic growth with respect to \xi …. We prove that the linear part of the limit problem is the homogenized matrix of the linear problem and the nonlinear part is given by H^0( u, Du) , where H^0 is defined by H^0( s,\xi) =\int_{]0,1[^n-\overline T} H ( y,s,C(y)\xi)\, {\rm d}y \quad \forall( s,\xi)\in\NBbbR\times\NBbbR^n , C( {\cdot/\varepsilon}) being the corrector matrices of the linear problem and T the reference hole. Show more

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 223-243, 1998

Authors: Shubov, Marianna A.

Article Type: Research Article

Abstract: We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordiantes from the 3‐dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. The nonconservative boundary conditions are given on the sphere |x|=a . Our main objects of interest are the nonselfadjoint operators in the energy space of 2‐component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and boundary conditions. Our main results are precise asymptotic formulas for the complex eigenvalues and eigenfunctions of these operators and the corresponding nonselfadjoint quadratic operator pencils. Based …on the asymptotic results of the present work, we will show in a forthcoming paper that the sets of root vectors of the above operators and of the dynamics generator corresponding to the full 3‐dimensional damped wave equation form Riesz bases in the appropriate energy spaces. Therefore, we will show that all these operators are spectral in the sense of Dunford. Show more

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 245-272, 1998

Authors: Aganović, I. | Jurak, M. | Marušić‐Paloka, E. | Tutek, Z.

Article Type: Research Article

Abstract: The asymptotic behaviour, with respect to the small period, of the equilibrium displacements corresponding to the Koiter’s shell model of periodically perturbed plate is considered. In the limit as the period tends to zero a model describing the deformation of a wrinkled plate is obtained in the two‐scale form. The two‐scale problem can be decoupled; the longitudinal displacement satisfies the classical equations for longitudinal deformations of a plate, while the transverse displacement is a solution of a fourth‐order elliptic problem modeling the anisotropic plate. Corresponding convergence result with correctors is proved by use of two‐scale convergence method. In the case …of wavy plate the effective coefficients of anisotropic plate are explicitly computed. Show more

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 273-297, 1998

Authors: Ono, Kosuke

Article Type: Research Article

Abstract: We consider the global existence and asymptotic stability of solutions to the Cauchy problem for degenerate nonlinear wave equations of Kirchhoff type with a dissipative term in unbounded domain. We derive the sharp decay estimates of the solution and its derivatives. Moreover, we show that the solution has a lower decay estimate of some algebraic rate.

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 299-314, 1998

Authors: Miranville, A. | Piétrus, A. | Rakotoson, J.M.

Article Type: Research Article

Abstract: Our aim in this article is to study the existence of finite dimensional attractors for a class of generalized Cahn–Hilliard equations based on a microforce balance introduced by M. Gurtin (Physica D 92 (1996), 178–192). We first consider some general results on the existence of finite dimensional attractors, and then apply these results to the generalized Cahn–Hilliard equations.

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 315-345, 1998

Authors: Kaise, Hidehiro | Nagai, Hideo

Article Type: Research Article

Abstract: Bellman–Isaacs equation of ergodic type is studied in paricular case. A solution of the equation related to the principal eigenfunction of the Schrödinger operator is obtained as the limit of the solution of a Bellman–Isaacs equation of discounted type. Furthermore, a singular limit of the equation is studied in relation to semi‐classical analysis.

Citation: Asymptotic Analysis, vol. 16, no. 3-4, pp. 347-362, 1998