# Backward Heat Equation with Transversality Condition

I have a backward parabolic equation of the form:

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

s.t.

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

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

$$$$U_{t} = - U_{xx}$$$$

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

$$$$W(x,H) = g(x)$$$$

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

$$$$\nu = H - \eta$$$$

to obtain

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

s.t.

$$$$W(x,0) = g(x)$$$$

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