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

John Meyer, David Needham*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
128 Downloads (Pure)

Abstract

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 that maxi=1,…,N⁡(supx∈RN⁡|uxi (n)(x,T)|)≥α, whilst limn→∞⁡(inft∈[0,T]⁡(maxi=1,…,N⁡(supx∈RN⁡|uxi (n)(x,t)|)−Ft(fn,0,u(n))))=0.

Original languageEnglish
Pages (from-to)3345-3362
JournalJournal of Differential Equations
Volume265
Issue number8
Early online date14 Jun 2018
DOIs
Publication statusPublished - 15 Oct 2018

Keywords

  • semi-linear parabolic PDE
  • functional derivative estimate

Fingerprint

Dive into the research topics of 'On a L functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity'. Together they form a unique fingerprint.

Cite this