Supose the following PDE in $x$ and $t$:
$u_t+(t+1)u_x=u_{xx}-2$
I know that stationary state is the solution when $u_t$ (dependence of time is $0$). So, can I also consider $t=0$ (getting the problem $u_x=u_{xx}-2$)? I mean, if the definition of stationary state is related whit the independence of time, could I consider any time fixed? It seems a little strange. For example, if I have:
$u_t+(t-1)u_x=u_{xx}-2$
What time should I get? Note that $t=0$ and $t=1$ lead to really different equations, $u_x=u_{xx}-2$ and $0=u_{xx}-2$.
Many thanks.