This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3:
The elliptic regularization of the heat equation is the PDE $$ u_t - \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where $\epsilon > 0$ and $U_T = U \times (0,t]$. Show that $(*)$ is the Euler-Lagrange equation corresponding to an energy functional $I_\epsilon[w] := \iint_{U_T} L_\epsilon(Dw,w_t,w,x,t) \, dx \, dt$.
(Hint: Look for a Lagrangian with an exponential term involving $t$.)
In my previous question with a different EL equation, I had to find a Lagrangian in terms of only three variables (that is, find $L=L(Du,u,x)$).
Now for this problem, I have to find a Lagrangian that is based with five variables and epsilon (that is, find $L=L_\epsilon(Du,u_t,u,x,t)$).
When the hint says "look for a Lagrangian with an exponential term involving $t$, do they mean something like $$e^{-t}(|Du|^2 +\epsilon u_t)$$ or similar?
Now, I got confused again because, unlike my previous question, for this one I am asked to work with the time variable $t$ sin addition to the spatial variable $x$. If the exponential term involving $t$ is indeed $e^{-t}$ times something, then I'm assuming upon differentiation with respect to $t$, product rule will be used to find our way back to the given EL equation.
If any answerers have any ideas, a good hint works best for me.