Global attractor for the one dimensional wave equation with displacement dependent damping

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor.     

Inverse Spectral Problem for the Schrödinger Equation with an Additional Linear Potential

Theoretical and Mathematical Physics - We consider the one-dimensional Schrödinger equation with an additional linear potential on the whole axis and construct a transformation operator with a...     

The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition

We examine the Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition. The question is addressed of existence of a solution with the same asymptotics at infinity as the initial condition. We demonstrate that the method of the inverse scattering problem is applicable to this problem.     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Energy estimates for solutions of the mixed problem for linear second-order hyperbolic equations

A mixed problem for a linear second-order hyperbolic equation with antidissipation inside the domain and dissipation on a part of the boundary is considered. It is proved that for certain relations between the antidissipation inside the domain and the dissipation on the part of the boundary, the energy of the system exponentially decreases, whereas for sufficiently large antidissipation inside the domain the boundary dissipation has no effect on the energy of the system; in this case the energy remains unbounded. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 483–488, April, 1996.     

The inverse scattering problem for a perturbed difference Hill equation

We consider the inverse scattering problem for the difference analog of a perturbed Hill equation. The perturbation coefficients are recovered from the periodic coefficients and from the scattering data.