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I'm currently dealing with the heat equation, but am having some issues. In particular, the following:

Let $f(x,t)$ be a solution to the heat equation $\frac{\partial f}{\partial t} = k \frac{\partial^2 f}{\partial x^2}$, s.t. $k>0$, $f(x,0) = f_0(x)$ is continuous in $[0,1]$, and $f_0(0) = f_0(1)$. Given $\int_0^1f_0(x)dx = 0$, find and prove $$\lim_{t\to\infty}e^{kt}f(x,t),$$ and describe the convergence (pointwise/$L^2$/uniform).

I would greatly appreciate some help!

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There are many solutions satisfying those conditions. Something is missing. If a boundary value problem is considered it would be handy to state it in full. –  Andrew Apr 30 '12 at 7:04

1 Answer 1

up vote 3 down vote accepted
  1. Use separation of variables to get the solution in the form $$ f(x,t)=a_0+\sum_{n=1}^\infty e^{-\lambda_nt}(a_n\cos(2\pi\,n\,x)+b_n\sin(2\pi\,n\,x)). $$ (I leave to you to calculate $\lambda_n$.)
  2. Use the condition on $f_0$ to calculate $a_0$.
  3. Obtain a formula for $e^{kt}f(x,t)$.
  4. Let $t\to\infty$.
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