Parabolic maximum principle

Remarks

First assume the strict inequality $\partial_t u - \Delta u < 0$. At a maximum point $(x_0, t_0)$ in $\Omega_T$ one would have $\Delta u(x_0, t_0) \le 0$ and $\partial_t u(x_0, t_0) \ge 0$, the latter also when $t_0 = T$, which is a contradiction. In general apply this to $v = u + \varepsilon \abs{x}^2$, for which $\partial_t v - \Delta v \le -2n\varepsilon < 0$, and let $\varepsilon \to 0$. Applied to $u$ and $-u$, it shows that a solution of the heat equation on $\Omega_T$ is bounded by its values on the parabolic boundary. So the initial-boundary value problem has at most one solution. The strong form says more: if $\Omega$ is connected and the maximum is reached at a point $(x_0, t_0)$ of $\Omega_T$, then $u$ is constant on $\overline{\Omega} \times [0, t_0]$. Boundedness of $\Omega$ is needed. On $\R \times (0, T]$, Tychonoff's example is a nonzero solution of the heat equation with zero initial values, so its maximum is not controlled by the initial values. Without the side boundary, uniqueness only holds under growth conditions, such as $\abs{u(x, t)} \le A e^{a \abs{x}^2}$.

Going deeper

For a student

The principle is the main tool for uniqueness and comparison results for parabolic equations. It extends to operators $\partial_t u - \sum_{i,j} a_{ij} \partial_i \partial_j u - \sum_i b_i \partial_i u$ with $(a_{ij})$ positive definite. It is the time-dependent analogue of the maximum principle for harmonic and subharmonic functions. Its strong form follows from a mean value property on heat balls or from a parabolic Harnack inequality.