heat conduction problem - well posedness
 

I was wondering if the following problem is well posed:

T_t = (a^2) T_xx, T(x,t) = temperature, 0 < x < 1, t > 0.

T(x,0) = f(x), T(0,t) = g(t), T_x(0,t) = h(t).

Instead of the last condition on T_x, if I had T(1,t) = e(t), then it is well posed and has a unique solution. Because the temperature and heat flux is specified on one end, I am wondering how one gets a Sturm-Liouville problem in x, in this case.

Thanks for all the help.:)

Ok. I did find out the answer. It is shown by Lavrentiev and others that this is NOT well posed.