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1.
It has been observed13 that the propagation of acoustic waves in the region Ω0= ?2 × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown9 in the case of Dirichlet's boundary condition U = 0 on ?Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω0. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω0 ?B, where B is a smooth bounded domain with B??Ω0. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ? 0 on ? B, where x ′ = (x1, x2, 0) and n is the normal unit vector pointing into the interior B of ? B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω0 will be discussed in a subsequent paper.  相似文献   

2.
Let Ω be a local perturbation of the n-dimensional domain Ω0 = Ropf;n ? 1 × (0, π). In a previous paper8 we have introduced the notion of an admissible standing wave. We shall prove that the principle of limiting absorption holds for the Dirichlet problem of the reduced wave equation in Ω at ω ≥ 0 if Ω does not allow admissible standing waves with frequency ω. From Reference 8, this condition is satisfied for every ω ≥ 0 if Ω ≠ Ω0, and v · x ′ ≤ 0 on δΩ, where x′ = ( x 1,…, xn ? 1, 0) and v is the normal unit vector on δΩ pointing into the complement of Ω. In contrast to this, the principle of limiting absorption is violated in the case of the unperturbed domain Ω0 at the frequencies ω = 1,2,… if n ≤ 3. The second part of our investigation, which will appear in a subsequent paper, is devoted to the principle of limit amplitude.  相似文献   

3.
In part 1
  • 1 Math. meth. in the Appl. Sci, 10, 125–144 (1988).
  • we studied the principle of limiting absorption for local perturbations Ω of the n-dimensional domain Ω0 = ?n?1 × (0, π). In this second part we extend our investigations to the time-dependent theory and show that absence of admissible standing waves implies the validity of the principle of limiting amplitude for every frequency ω≥0 if n ≠ 3 and for ω ≠ 2, 3,… if n = 3, respectively. In particular, the principle of limiting amplitude holds for every ω≥0 in the case n ≠ 3 and for every ω ≠ 2, 3,… in the case n = 3 if Ω≠Ω0 and ν · x ′ ?0 on ?Ω, where x ′ = (x1,…, xn?1, 0) and ν is the normal unit vector on ?Ω pointing into the complement of Ω This result stands in remarkable contrast to the fact that both principles are violated in the case of the unperturbed domain Ω0 at the frequencies ω = 1, 2,… if n?3. The question of the asymptotic behaviour of the solution as t→∞ for n = 3 and ω = 2, 3,… will be discussed in two subsequent papers.  相似文献   

    4.
    We consider the following one‐phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and with QT = ?n × (0, T). Under the condition that Ωo is convex and log uo is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ tT as long as ?Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth‐up‐to‐the‐interface solution, on 0 < t < Tc, with Tc denoting the extinction time of Ω(t). We also provide a new proof of a short‐time existence with C2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.  相似文献   

    5.
    The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. Ut ? 6uux + ?2uxxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved.  相似文献   

    6.
    The change of variable for the temperature Θ in the one-phase Stefan problem leads to the evolution inequality, (ut – Δuf)(vu) ? 0 for all regular v ? 0, where u ? 0 is required. This inequality is to hold over a space-time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ? Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {um}. The sequence {um} may be interpolated linearly in t to obtain an approximation UΔt of u. The following results are obtained in this paper: (i) a two-sided weak maximum principle for umum-1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of UΔt and ?UΔt, to u and ?u, respectively, on D ; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence xΔt, where xΔt interpolates x Δt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.  相似文献   

    7.
    The finite element (FE) solutions of a general elliptic equation ?div([aij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R 3, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω, so that the remaining part ΩB = Ω\Ω is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γa = Ω ∩ Ω B. In this article, instead of discarding an unbounded subdomain Ω and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006  相似文献   

    8.
    We present a global existence theorem for solutions of utt ? ?iaik (x)?ku + ut = ?(t, x, u, ut, ?u, ?ut, ?2u), u(t = 0) = u0, u(=0)=u1, u(t, x), t ? 0, x?Ω.Ω equals ?3 or Ω is an exterior domain in ?3 with smoothly bounded star-shaped complement. In the latter case the boundary condition u| = 0 will be studied. The main theorem is obtained for small data (u0, u1) under certain conditions on the coefficients aik. The Lp - Lq decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.  相似文献   

    9.
    10.
    We study positive solutions u to Δu + f(u) = 0 in Ω, u = 0 on ?Ω, and we address the following question: If Ω is a small perturbation of a ball, is u a small perturbation of a radially symmetric function? We prove two theorems which give an affirmative answer under different assumptions on the non-linearity f and on the topologies in which perturbations are considered.  相似文献   

    11.
    Let Ω denote an unbounded domain in ?n having the form Ω=?l×D with bounded cross‐section D??n?l, and let m∈? be fixed. This article considers solutions u to the scalar wave equation ?u(t,x) +(?Δ)mu(t,x) = f(x)e?iωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t→∞ is investigated. Depending on the choice of f ,ω and Ω, two cases occur: Either u shows resonance, which means that ∣u(t,x)∣→∞ as t→∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

    12.
    We exhibit new concentration phenomena for the equation ? ε2 Δu + u = up in a smooth bounded domain Ω ? ?2 and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter ε is converging to 0. For a suitable sequence εn → 0 we prove the existence of positive solutions un concentrating at the whole boundary of Ω or at some component. © 2002 Wiley Periodicals, Inc.  相似文献   

    13.
    In this paper we study the following problem: ut−Δu=−f(u) in Ω×(0, T)≡QT, ∂u ∂n=g(u) on ∂Ω×(0, T)≡ST, u(x, 0)=u0(x) in Ω , where Ω⊂ℝN is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u0(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper–lower solutions using related equations. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.  相似文献   

    14.
    《偏微分方程通讯》2013,38(7-8):1385-1408
    The purpose of this paper is to study the limit in L 1(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form ut ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.  相似文献   

    15.
    Let Ω be a bounded open and oriented connected subset of ? n which has a compact topological boundary Γ, let C be the Dirac operator in ? n , and let ?0,n be the Clifford algebra constructed over the quadratic space ? n . An ?0,n -valued smooth function f : Ω → ?0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F + ? F ? = f on Γ, where the components F ± are extendable to monogenic functions in Ω± with Ω+ := Ω, and Ω? := ? n \ (Ω ? Γ), respectively.  相似文献   

    16.
    The paper deals with the reconstruction of the convolution kernel, together with the solution, in a mixed linear evolution system of hyperbolic type. This problem describes uniaxial deformations u of a cylindrical domain (0,π) × Ω, which is filled with a linear viscoelastic solid whose material properties are supposed to be uniform on Ω‐sections perpendicular to the x axis. Various types of boundary conditions in [0,T] × {0,π} × Ω are prescribed, whereas Dirichlet conditions are assumed in [0,T] × (0,π) × Ω. To reconstruct both u and k, we suppose of knowing for any time tand any x ∈ (0,π) the flux of the viscoelastic stress vector through the boundary of the Ω‐section. The main novelty is that the unknown kernel k is allowed to depend, not only on the time variable t but also on the space variable x.  相似文献   

    17.
    18.
    For the Helmholtz equation Δu + k 2 u = 0 in a domain Ω with a cylindrical outlet Q + = ω × ?+ to infinity, we construct a fictitious scattering operator $\mathfrak{S}$ that is unitary in L 2(ω) and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in Ω and the subspace of eigenfunctions of $\mathfrak{S}$ corresponding to the eigenvalue 1 and orthogonal to the eigenfunctions with eigenvalues λnk 2 of the Dirichlet problem for the Laplace operator on the cross-section ω.  相似文献   

    19.
    We consider a domain Ω in ?n of the form Ω = ?l × Ω′ with bounded Ω′ ? ?n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)m + (? ? ?… ? ?)m]u = fe?iωt. We show that resonances can occur if 2ml. In particular, the amplitude of u may increase like tα (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.  相似文献   

    20.
    Given a bounded regular domain Ω in ℝN, we study existence and asymptotic behaviour of the solutions of the equation Δu + |Du|q = f(u) in Ω, which diverge on ∂Ω. We extend and complete some results contained in [4].  相似文献   

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