Mixed problem for the inhomogeneous wave equation with a summable potential

The resolvent approach to the Fourier method is used to examine the behavior of the formal solution to the mixed problem for an inhomogeneous wave equation with a summable potential. A classical solution is obtained under minimum conditions imposed on the initial function. It is shown that, in the case of square summable initial and driving functions, the series sum of the formal solution is a weak solution of the mixed problem.     

Second order elliptic boundary value problems in a domain with edges

The spectrum and essential spectrum of the Dirichlet problem for an arbitrary elliptic operator with real constant coefficients are studied,     

Mixed problem for the Petrovskii well-posed equation in a cylindrical domain

We study the problem of existence and uniqueness of the solution of a mixed problem for the Petrovskii well-posed equation in a cylindrical domain and the behavior of this solution for large values of time.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

Mixed problem for a semilinear ultraparabolic equation in an unbounded domain

We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation {fx1870-01} in an unbounded domain with respect to the variables x. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1661–1673, December, 2007.     

Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L ₂[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.     

Mixed finite element method for a strongly damped wave equation

A mixed finite element Galerkin method is analyzed for a strongly damped wave equation. Optimal error estimates in L²‐norm for the velocity and stress are derived using usual energy argument, while those for displacement are based on the nonstandard energy formulation of Baker. Both a semi‐discrete scheme and a second‐order implicit‐time discretization method are discussed, and it is shown that the results are valid for all t > 0. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 105–119, 2001     

Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an a priori estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 244–254, February, 1997.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝ ⁿ . For such problems, equivalent equations on the boundary in the simplest L ₂-spaces H ^s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _p ^s of Bessel potentials and Besov spaces B _p ^s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.     

Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables

We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to the space variables. The case of arbitrarily many space variables is considered. We establish conditions for the existence and uniqueness of a solution of this problem independent of the behavior of the solution as |x| → + ∞. The indicated classes of existence and uniqueness are the spaces of locally integrable functions, and, furthermore, the dimension of the domain does not limit the order of nonlinearity. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1523–1531, November, 2007.