Invariant interval for a cubic reaction-diffusion equation ($|u|\le \max (\Vert u_0{\Vert }_{\infty },\sqrt{\max (\lambda ,0)})$)

For $u_t=\Delta u+\lambda u-u^3$ with zero Dirichlet boundary condition, choose $K\ge \max (\Vert u_0{\Vert }_{\infty },\sqrt{\max (\lambda ,0)})$. The constants $K$ and $-K$ are a supersolution and a subsolution. More explicitly, testing with $(u-K{)}_{+}$ gives $\frac{1}{2}\frac{d}{dt}\Vert (u-K{)}_{+}{\Vert }_2^2\le \lambda \Vert (u-K{)}_{+}{\Vert }_2^2$, since the reaction derivative is at most $\lambda$ and the reaction at $K$ is nonpositive. The Gronwall inequality and the corresponding argument for $-u$ prove that $[-K,K]$ is invariant. A local variation-of-constants formula in the supremum norm then has a lifespan depending only on $K$, which prevents finite-time loss of the classical solution.