= Invariant interval for a cubic reaction-diffusion equation
{title2=$|u|\leq\max(\|u_0\|_\infty,\sqrt{\max(\lambda,0)})$}
For $u_t=\Delta u+\lambda u-u^3$ with zero <Dirichlet boundary condition>, choose $K\geq\max(\|u_0\|_\infty,\sqrt{\max(\lambda,0)})$. The constants $K$ and $-K$ are a supersolution and a subsolution. More explicitly, testing with $(u-K)_+$ gives $\frac12\frac d{dt}\|(u-K)_+\|_2^2\leq\lambda\|(u-K)_+\|_2^2$, 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>.
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