Uniform stabilization for the Timoshenko beam by a locally distributed damping.
In this note we prove the exponential decay of solutions of a quasilinear wave equation with linear damping and source terms.
This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining...
This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining...
We show that the differential inequality has the unique continuation property relative to the Sobolev space , , , if satisfies the conditionfor all compact , where if , we replace by . This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, , in the case . The proof uses Carleman’s approach together with the following pointwise inequality valid for all and any
We prove unique continuation for solutions of the inequality , a connected set contained in and is in the Morrey spaces , with and . These spaces include for (see [H], [BKRS]). If , the extra assumption of being small enough is needed.
Much of this paper will be concerned with the proof of the followingTheorem 1. Suppose d ≥ 3, r = max {d, (3d - 4)/2}. If V ∈ Llocr(Rd), then the differential inequality |Δu| ≤ V |∇u| has the strong unique continuation property in the following sense: If u belongs to the Sobolev space Wloc2,p and if |Δu| ≤ V |∇u| andlimR→0 R-N ∫|x| < R |∇u|p' = 0for all N then u is constant.
We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We perform measurements of current and voltage type on a (known) part of the boundary of the conductor. We prove that, even if the defects are unknown, the current and voltage measurements at the boundary uniquely determine the corresponding electrostatic potential inside the conductor. A corresponding stability result, related to the stability of Neumann problems with...
In this paper we prove a unique continuation result for a cascade system of parabolic equations, in which the solution of the first equation is (partially) used as a forcing term for the second equation. As a consequence we prove the existence of ε-insensitizing controls for some parabolic equations when the control region and the observability region do not intersect.
We study a non standard unique continuation property for the biharmonic spectral problem in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle , and , a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes...
We study a non standard unique continuation property for the biharmonic spectral problem in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle , and , a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing...
In this paper we present some results on the uniqueness and existence of a class of weak solutions (the so called BV solutions) of the Cauchy-Dirichlet problem associated to the doubly nonlinear diffusion equationb(u)t - div (|∇u - k(b(u))e|p-2 (∇u - k(b(u))e)) + g(x,u) = f(t,x).This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media, gases flowing in pipelines, etc. The solvability of this problem is established in the BVt(Q) space....