Suppose \( \Omega \subseteq \mathbb{R}^d \) is a domain, \( u \in C^{(2,1)}(\Omega_T) \) for some \( T > 0 \) with

\[ \partial_t u - \Delta u = 0. \]

Then for every heat ball \( E(x,t;r) \subseteq \Omega_T \), the following equality holds:

\[ u(x,t) = \frac{1}{4r^d} \iint_{E(x,t;r)} u(y,s) \frac{|x - y|^2}{(t - s)^2} \, dy\,ds. \]

[1, 2.3 Theorem 3]

Proof

Due to a transformation argument it is sufficient to consider \( (x,t) = (0,0) \) and \( E_r = E(0,0;r) \). Set

\[ \varphi(r) = \frac{1}{4r^d} \iint_{E_r} u(y,s) \frac{|y|^2}{s^2} \, dy\,ds. \]

Using partial integration we can show that \( \varphi' \equiv 0 \). Thus \( \varphi \) is constant and since

\[ \frac{1}{4r^d} \iint_{E_r} \frac{|y|^2}{s^2} \, dy\,ds = 1 \]

we obtain

\[ \varphi(r) = \lim_{s \to 0} \varphi(s) = u(0,0). \]
Remarks
  • \( u(x,t) \) depends only on values in the past.
  • Proved by Fuller (1966)

See also Link to heading

References Link to heading

  1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.