Suppose that \(U\subset \mathbb {R}^n\) is a bounded domain and \(b:U\to \mathbb {R}^n\) is a smooth velocity field. Suppose that \(u(x, t)\) solves the reaction-diffusion- advection equation with a logistic reaction term
\[ \begin {cases} \partial _t u+b(x)\cdot \nabla u-\Delta u=u(1-u) & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ). \end {cases} \]
Show that if \(u(x, 0) < 1\) in \(U\) then \(u(x, t) < 1\) for all \(t > 0\) and all \(x \in U\).