Problem For Second-order Hyperbolic Equation Research Articles

A time-nonlocal inverse problem for a hyperbolic equation with an integral overdetermination condition

This article is concerned with the study of the unique solvability of a timenonlocal inverse boundary value problem for second-order hyperbolic equation with an integral overdetermination condition. To study the solvability of the inverse problem, we first reduce the considered problem to an auxiliary system with trivial data and prove its equivalence (in a certain sense) to the original problem. Then using the Banach fixed point principle, the existence and uniqueness of a solution to this system is shown. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the inverse coefficient problem is proved for the smaller value of time.

Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients

The classical solution of the first mixed problem for a second-order hyperbolic equation with variable coefficients in the case of two independent variables in a curvilinear half-strip is considered. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. A method is proposed for constructing the solution using the method of sequential approximations for a system of integral equations of the second kind.

Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II:L2(Ω)-estimates

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C2-boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H1(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H1-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C2-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H2-solutions, and ultimately for H1-solutions. The passage from H2- to H1-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H1-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ0,Γ1} that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L2(Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L2(Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L2(Ω), by an attractive boundary feedback.

The mixed problem for a degenerate second-order hyperbolic equation

We study conditions for uniqueness and existence of a solution for the mixed problem for a second-order hyperbolic equation that is degenerate at a finite instant of time.

Smoothness of solutions of second-order hyperbolic equations with discontinuous coefficients

Diffraction problems for second-order hyperbolic equations were studied in [1-4] and other works. In those works, the solvability of the problems was established and the smoothness of the solutions was investigated near the surfaces of discontinuities of coefficients, except for the points of intersections of those surfaces with one another or with the boundary of the region. In the present paper, the smoothness of the generalized solution of a problem is studied in a neighborhood of the manifold of intersections of the discontinuity surfaces with the boundary of the region.

Mixed problem for second-order hyperbolic equation with first-order complex boundary condition

Mixed problem for second-order hyperbolic equation with first-order complex boundary condition

An approximation theorem for second-order evolution equations

A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos ? for ??0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.