I am asked to show that the transformation \begin{equation} u=-2\mu\partial_x\log\phi \end{equation} brings the viscous Burgers equation
\begin{equation} \partial_tu+u\partial_xu=\mu\partial_{xx}u \end{equation} to the heat equation \begin{equation} \partial_t\phi=\mu\partial_{xx}\phi \end{equation} given that \begin{equation} u\rightarrow0\text{ as }|x|\rightarrow\infty\end{equation}
I can show that the transformation gives me the equation \begin{equation} \partial_t\phi=\mu\partial_{xx}\phi -\frac{\phi}{2\mu}\psi(t)\end{equation}
where $\psi(t)$ is the 'constant' from integration over $x$, but I am struggling to get rid of that last term. Every other proof/answered question I have seen has finite boundary conditions so I am not really sure how to make use of this infinite boundary condition.