If a function $\phi(x,t)$ is $C^2$ and has compact support on $\mathbb{R}$ x $[0,T]$ does that mean that:
$\phi(x,T)=0$
$\phi(x,0)=0$
$\phi(\infty,t)=0$
$\phi(-\infty,t)=0$
as well as
$\phi_t(x,T)=0$
$\phi_t(x,0)=0$
$\phi_x(\infty,t)=0$
$\phi_x(-\infty,t)=0$
I am trying to check my understanding of this idea.