Answer by Denis Serre for Heat equation close to the steady state

Let $v:=\partial_xu$. It still satisfies the heat equation $\partial_tv=\partial_x^2v$, but with the Neumann boundary condition:$$\partial_xv(t,0)=\partial^2u(t,0)=\partial_tu(t,0)=\partial_t0=0,$$and the same at $x=1$. Therefore you may apply the maximum principle:$$\sup_x|v(t,x)|\le\sup_x|v(0,x)|=\sup_x|u_0'(x)|\le\delta.$$

Edit. To see why the maximum principle holds true, let me remind that $v$ is ${\cal C}^\infty$ up to the boundary for $t>0$. Then extending $V$ by parity to $x\in(-1,1)$, then by $2$-periodicity, one obtains a solution $V$ of the heat equation in ${\mathbb R}^+\times{\mathbb R}$. This one satisfies the maximum principle, here $|V|\le\delta$, which gives the result.