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In this paper, we consider a $L^\infty$ functional derivative estimate forthe first spatial derivative of bounded classical solutions $u:\mathbb{R}\times[0,T]\to\mathbb{R}$ to the Cauchy problem for scalar semi-linear parabolicpartial differential equations with a continuous nonlinearity$f:\mathbb{R}\to\mathbb{R}$ and initial data $u_0:\mathbb{R}\to\mathbb{R}$, ofthe form, \[ \sup_{x\in\mathbb{R}}|u_x (x , t)| \leq \mathcal{F}_t (f,u_0,u) \\ \ \forall t\in [0,T] . \] Here $\mathcal{F}_t:\mathcal{A}_t\to\mathbb{R}$ isa functional as defined in \textsection 1. We establish that the functionalderivative 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 dataand 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 \[\lim_{n\to\infty} \left( \inf_{t\in [0,T]} \left(\sup_{x\in\mathbb{R}}|u_x^{(n)}(\cdot , t)| - \mathcal{F}_t (f_n , 0 , u^{(n)})\right) \right) = 0 . \]

journal of differential equations

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