Blowup of Solutions and Boundary Instabilities in Nonlinear Hyperbolic Equations

Abstract

We construct elementary examples of systems of hyperbolic equations having solutions which blow up in finite time. We explicitly describe the system, initial data and solution. First, we exhibit a 3x3 system with compactly supported data which blows up in finite time. The solutions blows up in amplitude (L^infinity] norm) on an entire interval, so there is no possibilty of continuing the solution beyond the blowup time. We then consider a system of two Burger equations which are coupled through linear boundary conditions. We record the interesting observation that although the IBVP with a single boundary condition is globally well-posed, when two boundary conditiond are used on a finite domain, the IBVP is ill-posed. Because waves are reflected back into the domain, multiple interactions combine to give blowup in finite time, for arbitrarily small initial data. We conclude that some global integral or energy condition must be imposed in order to expect stability of solutions to IBVPs on compact domains. Finally, we show that the presence of shocks is not necessary, by exhibiting solutions which are continuous in the nonlinear fields. However, our solutions do contain discontinuities in the linearly degenerate field.