On a $L^\infty$ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

In this paper, we consider a $L^\infty$ functional derivative estimate for the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to the Cauchy problem for scalar semi-linear parabolic partial differential equations with a continuous nonlinearity $f:\mathbb{R}\to\mathbb{R}$ and initial data $u_0:\mathbb{R}\to\mathbb{R}$, of the form, $$\sup_{x\in\mathbb{R}}|u_x (x , t)| \leq \mathcal{F}_t (f,u_0,u) \ \ \ \forall t\in [0,T] .$$ Here $\mathcal{F}_t:\mathcal{A}_t\to\mathbb{R}$ is a functional as defined in \textsection 1. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence $(f_n,0,u^{(n)})$, where for each $n\in\mathbb{N}$, $u^{(n)}:\mathbb{R}\times [0,T]\to\mathbb{R}$ is a solution to the Cauchy problem with zero initial data and nonlinearity $f_n:\mathbb{R}\to\mathbb{R}$, and for which $\sup_{x\in\mathbb{R}} |u_x^{(n)}(x,T)| \geq \alpha >0$, with $$\lim_{n\to\infty} \left( \inf_{t\in [0,T]} \left( \sup_{x\in\mathbb{R}}|u_x^{(n)}(\cdot , t)| - \mathcal{F}_t (f_n , 0 , u^{(n)}) \right) \right) = 0 .$$