Estimate of the time of occurrence of discontinuities in the solution of a boundary value problem for a second order quasilinear hyperbolic system: PMM vol. 36, n≗3, 1972, pp. 528–532

Abstract

An estimate is obtained for the time of occurrence of a discontinuity in the solutions of a hyperbolic system of quasilinear differential equations with zero initial and continuous boundary conditions. Examples are presented for one-dimensional gasdynamics and geometrically nonlinear elasticity theory problems. Problems of the analysis of nonlinear transient waves in gasdynamics and elasticity theory result in the integration of quasilinear hyperbolic systems of differential equations. Problems of the occurrence of discontinuities in the solutions of such systems under continuous initial conditions (the Cauchy problem), as well as under continuous boundary conditions (the boundary value problem) have been examined in [1– 6], The most general result has been obtained in [4] for the Cauchy problem, for which upper and lower bounds for the time of occurrence of discontinuities in the solution of a second order system have been established by using a congruence theorem. An analogous result is obtained herein for the boundary value problem by using the Jeffrey method [4].