# Backward Heat Equation with Transversality Condition

I have a backward parabolic equation of the form:

$$\begin{equation} W_{\eta} + aW_{xx} - bW = 0 \end{equation}$$

s.t.

$$\begin{equation} \lim_{\eta \rightarrow \infty}(x,\eta) = g(x) \end{equation}$$

were $$x \in \mathbb{R}$$, $$\eta \geqslant 0$$, and $$a,b$$ are positive constants.

Applying the following transformations:

\begin{align} W(x,\eta) &= U(x,t)e^{b\eta} \\ t &= a\eta \end{align}

we would get the backward heat equation below

$$\begin{equation} U_{t} = - U_{xx} \end{equation}$$

However, the transversality condition becomes a problem, since as $$\eta \rightarrow \infty$$, $$e^{b\eta} \rightarrow \infty$$.

Usually, if the terminal condition is of the form

$$\begin{equation} W(x,H) = g(x) \end{equation}$$

with $$H$$ finite, we could "reverse" it, that is, we could apply the following transformation:

$$\begin{equation} \nu = H - \eta \end{equation}$$

to obtain

$$\begin{equation} -W_{\nu} + aW_{xx} - bW = 0 \end{equation}$$

s.t.

$$\begin{equation} W(x,0) = g(x) \end{equation}$$

which we can solve the traditional way (Fourier transform). However, as my terminal condition happens only at infinity I can't apply the reverse transformation above, thus I don't know how to overcome this problem. Any hint or reference?

• Have you tried solving the equation with $U(x,T)=g(x)$ and let $T\to\infty$? – Dylan Apr 2 at 7:23
• I have edited the question, now it reflects properly the problem I have. – Nicolas Pimentel de Souza Apr 2 at 14:48