given the 2 PDE
$$ \Delta u-au_{tt}+u_{t}=0$$
and $$ \Delta u + Du*Df=0 $$
here $ \Delta u $ is the Laplacian $ Du= grau$ and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$
my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks