On a L^∞ 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^∞ functional derivative estimate for the first spatial derivatives of bounded classical solutions u:R^N×[0,T]→R to the Cauchy problem for scalar second order semi-linear parabolic partial differential equations with a continuous nonlinearity f:R→R and initial data u₀:R^N→R, of the form, maxi=1,…,N⁡(supx∈R^N⁡|u_xi (x,t)|)≤F_t(f,u₀,u)∀t∈[0,T]. Here F_t:A_t→R is a functional as defined in §1 and x=(x₁,x₂,…,x_n)∈R^N. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence (f_n,0,u⁽ⁿ⁾), where for each n∈N, u⁽ⁿ⁾:R^N×[0,T]→R is a solution to the Cauchy problem with zero initial data and nonlinearity f_n:R→R, and for which there exists α>0 such that maxi=1,…,N⁡(supx∈R^N⁡|u_xi ⁽ⁿ⁾(x,T)|)≥α, whilst limn→∞⁡(inft∈[0,T]⁡(maxi=1,…,N⁡(supx∈R^N⁡|u_xi ⁽ⁿ⁾(x,t)|)−F_t(f_n,0,u⁽ⁿ⁾)))=0.