posted by Anonymous .
Not sure how to approach this problem.
Prove the comparison principle for the diffusion equation, Ut = K*Uxx. If u and v are solutions, and if u <= v for t = 0, x = 0, and x=l, then u <= v for 0 <= t < infinity, and 0 <= x <= l
If you graph it for a constant positive Uxx (a parabola opening up) and zeros at x = 0 and L for u and above that for v you can see that u will remain less than v as time goes on and the temperature for example U flattens out.
Proving it is quite another matter though and I do not know how off hand