I have a question regarding some things I've seen in previous exams and couldn't find any justification for it.
Let us a look at the following problem: $$\left\{ \matrix{ {u_t}\left( {x,t} \right) = {u_{xx}}\left( {x,t} \right) \hfill \cr u\left( {x,0} \right) = f\left( x \right) \hfill \cr {u_x}\left( {0,t} \right) = g\left( t \right) \hfill \cr {u_x}\left( {L,t} \right) = h\left( t \right) \hfill \cr 0 < x < L{\rm{ }},{\rm{ }}t > 0 \hfill \cr} \right.$$
For the sake of simplicity let us assume the functions $f,g,h$ are infinitely differentiable on the real line.
In many exam exercises I've encountered, the following line boggled me, since no justification for it was provided: $$\mathop {\lim }\limits_{t \to \infty } {d \over {dx}}u\left( {x,t} \right) = {d \over {dx}}\mathop {\lim }\limits_{t \to \infty } u\left( {x,t} \right)$$ It is rather well known that interchanging limit and derivative is not always allowed.
But since I've encountered this numerous times and every time it was treated as something "trivial" in the official solutions provided by my professor.
So am I missing some theorem regarding the solutions of the heat equation and the ability to interchange limit and derivative?
Set $ u_n (x) = u (x, n) $, where $ n \in \mathbb {N} $
You are essentially asking whether you could differentiate the limit $\lim_{n \to \infty} u_n(x)$ and what is it equal to.
This might help :
This is a stronger version (uniformly integrability condition) of what you need:
If a sequence of absolutely continuous functions {$f_n$} converges pointwise to some $f$ and if the sequence of derivatives {$f_n’$} converges almost everywhere to some $g$ and if {$f_n’$} is uniformly integrable then $\lim\limits_{n\mapsto \infty} f_n’ = g= f’$ almost everywhere. Where the derivative of $f$ is $f’$. If the convergence is pointwise and $ g $ is continuous then $ f'$ = $ g $ everywhere.
Proof : by FTC $f_n(x) – f_n(a) = \int_a^x f_n’ dx$
By Vitali convergence theorem : $\lim\limits_{n\mapsto \infty}\int_a^x f_n’ dx = \int_a^x g dx$
Therefore $\lim\limits_{n\mapsto \infty}( f_n(x) – f_n(a))= \int_a^x g dx$
$f(x)-f(a) = \int_a^x g dx$
$f(x)’=g$ almost everywhere
If the convergence is pointwise and $ g $ is continuous then $ f'$ = $ g $ everywhere.