This question comes from the text book Partial Differential Equations An Introduction, Walter A. Strauss. I was trying to do some of text book questions as I am doing a course in PDE's and had a bit of trouble with one of the questions.
Question
A homogeneous body occupying a solid region D is completely insulated. Its initial temperature is f(x). Find the steady state temperature that it reaches after a long time. (Hint: No heat is gained or lost)
Solution
So u know that the heat equation is given by
$$ u_t = ku_{xx}$$
where $k\neq0$. Since we know the the temperature is in steady state (i.e. there is no heat lost or gained) then
$$ u_t= 0 $$
substituting this back into our PDE we find
$$ 0 = ku_{xx} \implies 0 = u_{xx} .$$
Since the body is completely insulated we know
$$ 0 = u_{x} $$
Finally we can can say
$$ u = constant $$but i am a bit stuck on how the find the steady state temperature now that I have the general solution to u ? Any help would be appreciated!