# Finite and infinite speed of propagation for wave and heat equation

What is the formal definition of Finite and infinite speed of propagation?

I have searched for it, is the finite one means the solution is only determined by a bounded region?

Also I do not understand the meaning of its name finite "speed of propagation".

I understand the word finite but what is the meaning of speed of propagation?

The speed of propagation is discussed in the context of initial value problems: we prescribe initial condition $$u(x,0)=\phi(x),\quad x\in\mathbb{R}^n$$ possibly also for derivatives: $u_t(x,0)=\psi(x)$, etc.
If the initial data consists of functions with compact support, then for every $t>0$ the solution $u(\cdot,t)$ has compact support.
One can make this quantitative: the speed of propagation is $\le c$ provided that the following holds:
If the initial data consists of functions with support contained in a ball $B(a,R)$, then for every $t>0$ the solution $u(\cdot,t)$ has support contained in $B(a,R+ct)$.
Finally, the infimum of $c$ for which the above holds could be taken as the speed of propagation.