Is every positive ancient solution of heat equation to be constant?

Question

Let $u$ be a positive solution of heat equation: $$ \partial _tu(x,t)=\Delta u(x,t) $$ on $\mathbb R^n \times(-\infty,0)$.

Then must $u$ be a constant?

Answer 1

Try $u(x,t) = \exp(|k|^2 t + k \cdot x)$