Finite speed of propagation

Remarks

The proof uses the energy method on shrinking balls. Let $e(t) = \frac{1}{2}\int_{B(x_0, t_0 - t)} \big(u_t^2 + \abs{\nabla u}^2\big)\, dx$ for $0 \le t \le t_0$. Differentiating, integrating by parts and using $\abs{u_t\, \partial_\nu u} \le \frac{1}{2}(u_t^2 + \abs{\nabla u}^2)$ on the boundary sphere gives $e'(t) \le 0$. Since $e(0) = 0$, it follows that $e \equiv 0$, so $u_t$ and $\nabla u$ vanish in the cone and $u$ equals its initial value $0$ there. For the equation $u_{tt} = c^2 \Delta u$, the same holds with the cone $\abs{x - x_0} \le c(t_0 - t)$. Example: in one dimension, d'Alembert's formula shows directly that $u(x, t)$ depends only on the data on $[x - t, x + t]$. By linearity, two solutions whose data agree on $B(x_0, t_0)$ agree in the cone, so a change of the data travels with speed at most $1$. The heat equation has no such property: a nonnegative nonzero initial temperature with compact support gives a positive temperature everywhere at every positive time.

Going deeper

For a student

Finite speed of propagation is the characteristic feature of hyperbolic equations, in contrast with parabolic and elliptic ones. It gives the domain of dependence of a point, the backward cone, and the domain of influence of a region of data, the forward cone. The energy proof works for general second-order hyperbolic equations with variable coefficients, with the cone replaced by a region bounded by characteristic surfaces. It also proves uniqueness for the Cauchy problem and allows one to solve the problem locally and patch the local solutions together.