A dramatic example of an ill-posed, second order hyperbolic PDE problem is given by the following BVP for the one dimensional wave equation.
It can be shown that if is irrational, then the only solution of this BVP for the wave equation is identically zero; whereas if is rational, the problem has infinitely many nontrivial solution. Thus the solution fails to depend continuously on the data – namely on the size of the region on which the problem is stated.
Following in another example. Let us consider the solutions of the vibrating string equation
in the rectangle
satisfying the conditions
where is a given constant in and is a given smooth function. We notice that, if is an irrational number, then there exists one and only one smooth solution of the problem under suitable assumptions on .
This is because a function satisfying the problem can be represented by a uniformly convergent series
and we have
Since , then which claims .
If is rational, the uniqueness does not hold. For example, if is an integer, called then the function satisfies the first two boundary conditions and
We also observe that the problem is not well posed. In fact, small changes of datum in -norm ( any integer) lead to arbitrarily large changes of solution in -norm.
The stability of the problem can be restored by an additional condition, for example by an a priori bound on the energy, i.e.
where is a given constant.