On a $L^{\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$ functional derivative estimate for the first spatial derivative of bounded classical solutions $u:$\mathbb{R}$\times [0,T]\to$\mathbb{R}$$ to the Cauchy problem for scalar semi-linear parabolic partial differential equations with a continuous nonlinearity $f:\mathbb{R}\to \mathbb{R}$ and initial data $u_0:\mathbb{R}\to \mathbb{R}$, of the form, $$\underset{x\in \mathbb{R}}{\sup }\mathrm{\mid }{u}_x(x,t)\mathrm{\mid }\le {\mathcal{F}}_t(f,u_0,u)\mathrm{\forall }t\in [0,T]\mathrm{.}$$ Here ${\mathcal{F}}_t:{\mathcal{A}}_t\to \mathbb{R}$ is a functional as defined in \textsection 1. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence $(f_n,0,u^{(n)})$, where for each $n\in \mathbb{N}$, $u^{(n)}:$\mathbb{R}$\times [0,T]\to$\mathbb{R}$$ is a solution to the Cauchy problem with zero initial data and nonlinearity $f_n:\mathbb{R}\to \mathbb{R}$, and for which ${\sup }_{x\in \mathbb{R}}\mathrm{\mid }{u}_x^{(n)}(x,T)\mathrm{\mid }\ge \alpha >0$, with $$\underset{n\to \mathrm{\infty }}{\lim }\left(\underset{t\in [0,T]}{\inf }(\underset{x\in \mathbb{R}}{\sup }\mathrm{\mid }{u}_x^{(n)}(\cdot ,t)\mathrm{\mid }-{\mathcal{F}}_t(f_n,0,u^{(n)}))\right)=0.$$