Consider the wave equation with initial data:
Hadamard showed that this problem is ill-posed: there exist large solutions with arbitrarily small initial data. For instance, if we take , then and , then we can make u(t,x) grow arbitrarily fast while keeping and small.
Tweaking this construction, it is not hard to see that for any given k and , we can construct initial data such that
and This can be interpreted as saying that the problem is ill-posed even in a Holder sense.
My question is: can one construct an example of a solution u(t,x) with initial data and such that