Stability of the solution to a boundary value problem in hydrodynamics (Q1360809)

The aim of the paper is to study the stability of the zero solution of the operator equation: \[ \ddot u(t)(B_1(t)+ B_2(t))\dot u(t)+ (A_1(t)+ A_2(t))u(t)=0\tag{1} \] with the initial conditions \[ u(0)= u_0,\quad \dot u(0)= u_1.\tag{2} \] It is assumed that the operators \(A_1(t)\), \(A_2(t)\), \(B_1(t)\), and \(B_2(t)\) are defined on a constant domain \(D_0\subset H\) (\(H\) is a Hilbert space) and have the following properties: 1) \(A_1(t)\) is selfadjoint and strongly continuous in \(t\); \(B_1(t)\) is symmetric; \(A_2(t)\) and \(B_2(t)\) are skew-symmetric; 2) There exists a \(p>0\) such that \(|A_1(t)|\geq pI\) for all \(t\); 3) There exists a \(b_1>0\) such that \(B_1(t)\geq b_1I\) for all \(t\); 4) For all \(t\) we have \(D_0\subset D_t'\), where \(D_t'\) is the domain of \(\dot A(t)\), and \[ \dot A(t)v= s-\lim_{\Delta t\to 0} {A(t+\Delta t)- A(t)\over\Delta t} v. \] There are given conditions on the operators \(A_1(t)\), \(A_2(t)\), \(B_1(t)\), and \(B_2(t)\) sufficient for exponential stability, instability, and stability of the zero solution of the problems (1), (2). Later these results are applied to the problems \[ {\partial^2u\over\partial t^2}+ \Biggl(\alpha {\partial^5u\over\partial x^4\partial t}+ k {\partial u\over\partial t}+ 2\beta U(t) {\partial^2u\over\partial x\partial t}\Biggr)+ \Biggl({\partial^4u\over\partial x^4}+ U^2(t) {\partial^2u\over\partial x^2}+\beta {\partial^2U(t)\over\partial t} {\partial u\over\partial x}\Biggr)= 0,\tag{3} \] \[ u(0,t)= u(1,t)= {\partial u\over\partial x} (0,t)={\partial u\over\partial x} (1,t)=0,\tag{4} \] which describes the vibrations of a pipeline (with fixed ends) placed in a viscous medium and carrying the flow of an incompressible fluid.