Let \(g \in C_b(\mathbb{R}^d)\) and \(u \in C^{(2,1)}\left(\mathbb{R}^d \times (0,T)\right) \cap C^0\left(\mathbb{R}^d \times [0,T]\right)\) be a solution of

\[ \begin{cases} \partial_t u - \Delta u = 0 & \text{in } \mathbb{R}^d \times (0,T), \\ u = g & \text{on } \mathbb{R}^d \times \{t=0\}. \end{cases} \]

If there are \(a, A > 0\) such that \(u(x,t) \leq A e^{a\lvert x\rvert^2}\) on \(\mathbb{R}^d \times [0,T]\), then

\[ \sup_{\mathbb{R}^d \times [0,T]} u = \sup_{\mathbb{R}^d} g. \]

In particular, this implies that under this side condition solutions are unique.

Proof

Find \(v\) such that \(v\) is a solution with \(v(x,t) \leq u(x,0) \leq \sup g\) on a small time interval. Apply the maximum principle.

Then, use a chain argument to get the full result.

Remarks
  • The statement is also true for \(T = \infty\).
  • One cannot apply the properties of the solution given in (0x683e72eb) because \(u\) is an arbitrary solution.
  • The nonhomogeneous problem has a unique solution if \(u\) satisfies the growth condition.
  • This result is proved by Tychonoff in 1935.
  • There are other solutions violating the condition above.