Toronto Math Forum
APM3462012 => APM346 Math => Misc Math => Topic started by: Thomas Nutz on October 05, 2012, 03:57:42 PM

we are given
$$
U(x,t)=\int_{\infty}^xu(x,t)dx
$$
and are to prove that $U$ satisfies $U_t=kU_{xx}$.
The proof given is "one can see easily that as $x\rightarrow \infty$ and that therefore $U$ and all its derivatives have to be zero". But the integral over any function with the upper bound approaching the lower bound goes to zero!
For instance I take the function $f(x)=x^5$, which obviously does not satisfy the heat equation for $x\neq 0$. Then isn't
$$lim_{x\rightarrow \infty}\int_{\infty}^{x}x^5dx=0$$, and according to this "proof" $\int_{\infty}^{x}x^5dx$ would satisfy the heat equation?
This does not make sense to me...
Thanks!

we are given
$$
U(x,t)=\int_{\infty}^xu(x,t)dx
$$
and are to prove that $U$ satisfies $U_t=kU_{xx}$.
The proof given is "one can see easily that as $x\rightarrow \infty$ and that therefore $U$ and all its derivatives have to be zero". But the integral over any function with the upper bound approaching the lower bound goes to zero!
Thanks!
First, a correct citation:
"as $x\to \infty $ $U$ is fast decaying with all its derivatives".
So, this is true for $R= U_ukU_{xx}$, right? But we know that $R$ does not depend on $x$ (fact, your 'counterexample' misses) and therefore $R=0$