The initial boundary value problem for a viscoelastic equation with nonlinear damping in a bounded domain is considered. By modifying the method, which is put forward by Li, Tasi and Vitillaro, we sententiously proved that, under certain conditions, any solution blows up in finite time. The estimates of the life-span of solutions are also given. We generalize some earlier results concerning this equation.
We focus on the blow-up in finite time of weak solutions to the wave equation with interior and boundary nonlinear sources and dissipations. Our central interest is the relationship of the sources and damping terms to the behavior of solutions. We prove that under specific conditions relating the sources and the dissipations (namely p > m and k > m), weak solutions blow up in finite time.
In this paper, we show finite time blow-up of solutions of the p−wave equation in ℝN, with critical Sobolev exponent. Our work extends a result by Galaktionov and Pohozaev [4]
We consider space semi-discretizations of the 1-d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a...
We study a model describing vibrations of a cylindrical domain with thickness e > 0. A characteristic of this model is that it contains pollution terms in the boundary data and missing terms in the initial data. The method of sentinels'' of J. L. Lions [7] is followed to construct a sentinel using the observed vibrations on the boundary. Such a sentinel, by construction, provides information on pollution terms independent of missing terms. This requires resolution of initial-boundary value...
En utilisant une méthode dépendante du temps, nous démontrons la complétude asymptotique pour l'équation des ondes dans une classe d'espaces-temps stationnaires et asymptotiquement plats. On introduit l'observable de vitesse asymptotique et on décrit son spectre (sous des hypothèses plus faibles que pour la complétude asymptotique). Les méthodes utilisées sont inspirées par celles de l'analyse du problème à deux corps en mécanique quantique.
A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.
We consider the controllability and observation problem for a simple model describing the interaction between a fluid and a beam. For this model, microlocal propagation of singularities proves that the space of controlled functions is smaller that the energy space. We use spectral properties and an explicit construction of biorthogonal sequences to show that analytic functions can be controlled within finite time. We also give an estimate for this time, related to the amount of analyticity of the...
We study boundary control problems for the wave, heat, and Schrödinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. Exact controllability in L₂-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. Null controllability for the heat equation...
In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces.
In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces.
In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces.