The stability of the solutions of some boundary-value problems for hyperbolic equations
The authors investigate the behaviour of the solutions of linear hyperbolic equations as \(t\to\infty\) in the half space \(x>0\), \(y_ i\in\mathbb{R}\), \(i=1,2,\dots, r\), with boundary conditions given for \(x=0\). The equation and boundary conditions are assumed to be homogeneous with respect to the order of differentiation and all coefficients are assumed to be constant. It is shown, that the disturbances may have the following type of behaviour as \(t\to\infty\): exponentially increasing (instability), decay as a power function (stability) and remains bounded (neutral stability). The transitions of the system to an unstable, stable and neutrally stable state are investigated and the criteria for these transitions are given. These criteria are used to establish the existence of neutrally stable magnetohydrodynamic shocks even in the case of an ideal gas, a phenomenon that has not been previously documented. The existence of an a priori bound on the solution has been proved for these systems in cases of stability and neutral stability. The interaction of disturbances with the boundary in the case of neutral stability produces a non-smooth solution, so that the a priori bound is unimprovable. The behaviour of disturbances on the boundary is described by the solution of the Cauchy problem for some systems of partial differential equations of a high order with special conditions on the external forces and the initial values.
scientific article; zbMATH DE number 500865