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?