Nikolai A. Larkin

Uniform Stabilization for a Hyperbolic Equation with Acoustic Boundary Conditions in Simple Connected Domains

We prove the existence and uniqueness of strong and weak solutions as well as the uniform stabilization of the energy of initial boundary value problem for a hyperbolic equation in a class of domains Ω ⊂ \( \Omega \subset \mathbb{R}^N \) which includes simply connected regions. The boundary Γ of Ω can be a smooth simple connected manifold and the boundary conditions are acoustic boundary conditions on a portion Γ1 of the boundary and the Dirichlet boundary condition on the rest of Γ.

Exponential decay for the linear Zakharov–Kuznetsov equation without critical domain restrictions

Abstract Initial–boundary-value problems for the linear Zakharov–Kuznetsov equation posed on bounded rectangles are considered. The spectral properties of a stationary operator are studied in order to show that the evolution problem posed on a bounded rectangle has no critical restrictions on its size. The exponential decay of regular solutions is established.

The 2D Kawahara Equation on a Half-Strip

We formulate on a half-strip an initial boundary value problem for the two-dimensional Kawahara equation. Existence and uniqueness of a regular solution as well as the exponential decay rate for the elevated norm

Initial-boundary value problem for a compressible dusty gas model

\begin{aligned} \Vert u\Vert ^2_{H^1(D)}(t)+\Vert u_{xx}\Vert ^2_{L^2(D)}(t) \end{aligned}

Regular solutions of the 2D Zakharov–Kuznetsov equation on a half-strip

‖u‖H1(D)2(t)+‖uxx‖L2(D)2(t)of small solutions as

Kawahara equation in a bounded domain

t \rightarrow \infty