The stability of the solutions of some boundary-value problems for hyperbolic equations

Abstract

The behaviour of the solutions of linear hyperbolic equations is investigated as t→ ∞ in the half-space x > 0, - ∞ < y α < ∞, α = 1,2, …, r, with boundary conditions defined on the boundary x = 0. The equations and the boundary condition are assumed to be homogeneous with respect to the order of differentiation and all coefficients are assumed to be constant. A problem of this type has been previously studied in detail in connection with the stability of shock waves in gas dynamics [1–4] and some particular results have also been obtained for magnetohydrodynamic shocks [5–7]. In general, as will be shown below, the disturbances may have the same types of behaviour as t→∞ as in [2,4]: the disturbances increase exponentially (instability), decay as a power function (stability), or remain 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 derived. 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 [5–7]. The existence of an a priori bound on the solution has been proved for these systems in cases of stability and neutral stability [8,9]. The interaction of disturbances with the boundary in the case of neutral stability produces a non-smooth solution, so that the a priori bound of [9] is unimprovable. An elementary explanation of this effect is proposed. It is shown that the addition of small non-differential terms to the equations and the boundary conditions does not cause the problems to become ill-posed if the parameters of the original problem ensure neutral stability. The behaviour of disturbances on the boundary of the half-space 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. This result is similar to that observed in gas dynamics [10]. The stability of solutions with boundary conditions at x = 0 for x>0 and x<0 is analysed similarly and does not require a separate treatment.