1
$\begingroup$

If we have the heat equation $u_t-ku_{xx}=0$, where $(x,t)\in\Re\times(0,\infty)$, $k>0$, and $u(x,0)=f(x)$, then the general solution is

\begin{align} u(x,t)=\int_{-\infty}^{\infty}\Phi(x-y,t)f(y)dy\,, \end{align}

where

\begin{align} \Phi(x,t)=\frac{1}{\sqrt{4\pi kt}}e^{-x^2/4kt}\,. \end{align}

If we have the same conditions, what would be the general solution to the heat equation for $k<0$?

I tried the change of variables, $x=ip$, where $p$ is pure imaginary, such that the heat equation became $u_t+ku_{pp}=0$. I am not sure how to solve this, though. I do know that I cannot assume that the spatial and temporal components of the equation are separable. I was thinking that the solution might be

\begin{align} u(p,t)=\int_{-i\infty}^{i\infty}\Phi(ip-iy,t)f(iy)dy\,, \end{align}

This satisfies $u_t+ku_{pp}=0$, but I am not sure if I am missing something crucial.

  • 1
    I think the solution diverges and it's not physically possible (where is all the energy coming from? Infinite heat?). – 2017-02-27
  • 0
    Ultimately, I'm trying to evaluate $e^{-t\partial_x^2}g(x)$. By setting this equal to $u(x)$ and differentiating both sides with respect to $t$, I get the backwards heat equation. I'm treating $t$ as a variable, though it is really a constant in my exponential, so the backwards heat equation is only a means to the end. What I want in the end is something like $u(x,0.03)$. Even though $u$ diverges as $t\to\infty$, does there exist a (divergent) solution? – 2017-02-27
  • 0
    This is the same as reversing time. Because of the smoothing that takes place in forward time, there may be not be a solution in the backwards time direction for more than a short period of time, or perhaps not at all( ?). – 2017-02-28
  • 0
    The backwards heat equation is not a very well-posed problem. Therefore, we cannot guarantee any degree of existence, uniqueness, or stability (we're very much concerned more with the stability condition). – 2017-03-01

0 Answers 0