Partial differential equations Parabolic equations Theorem

Widder's theorem

Statement

Let $n \ge 1$ and $T > 0$ . Let $u \colon \R^n \times [0, T) \to \R$ be continuous, with $\partial_t u$ and the second partial derivatives in $x$ continuous on $\R^n \times (0, T)$ . Assume that $u$ solves the heat equation $\partial_t u = \Delta u$ on $\R^n \times (0, T)$ and that $u \ge 0$ . Let $g(x) = u(x, 0)$ . Then for all $x \in \R^n$ and $0 < t < T$ , $u(x, t) = \int_{\R^n} \Phi(x - y, t) \, g(y) \, dy,$ where $\Phi$ is the heat kernel of $\R^n$ .

Remarks

In particular, a nonnegative solution with initial value $0$ is identically $0$ , so nonnegative solutions of the Cauchy problem are unique. No growth condition at infinity is needed, which is the point of the theorem. The sign condition cannot be dropped: Tychonoff's example is a nonzero solution with initial value $0$ , and it changes sign. One half of the proof is a comparison on large balls with the maximum principle, which gives $u \ge \int \Phi(x - y, t) g(y) \, dy$ . The main work is to show that the difference, a nonnegative solution with initial value $0$ , vanishes. David Widder proved the theorem for $n = 1$ ; the version stated holds in every dimension.

Going deeper

For a student

Without a sign condition, uniqueness for the Cauchy problem needs a growth bound such as $\abs{u} \le A e^{a \abs{x}^2}$ . Widder's theorem replaces this bound by positivity, which is often easier to check in applications, for example for probability densities. It is related to the parabolic Harnack inequality for nonnegative solutions.

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Heat equation concept
Heat kernel concept
Widder's theorem statement

Tag	`M0J1YJAE00`
URI	`m2://M0J1YJAE00`
Kind	statement, theorem
Writing	W0 stub
MSC	35K05, 35K15, 35B09
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About	Heat equation
Uses	Heat equation, Heat kernel
Changed	2 November 2026