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:RN×[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:RN→R, of the form,maxi=1,…,N⁡(supx∈RN⁡|uxi(x,t)|)≤Ft(f,u0,u)∀t∈[0,T]. Here Ft:At→R is a functional as defined in §1 and x=(x1,x2,…,xn)∈RN. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence (fn,0,u(n)), where for each n∈N, u(n):RN×[0,T]→R is a solution to the Cauchy problem with zero initial data and nonlinearity fn:R→R, and for which there exists α>0 such thatmaxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,T)|)≥α, whilstlimn→∞⁡(inft∈[0,T]⁡(maxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,t)|)−Ft(fn,0,u(n))))=0.