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 (Linfinity] 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.