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

• Published in: arXiv.org
• Main Authors: Meyer, John Christopher, Needham, David John
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.07.2016
• Subjects: Cauchy problems; Mathematical analysis; Nonlinearity; Partial differential equations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1607.07102

In this paper, we consider a \(L^\infty\) 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, \[ \sup_{x\in\mathbb{R}}|u_x (x , t)| \leq \mathcal{F}_t (f,u_0,u) \ \ \ \forall t\in [0,T] . \] Here \(\mathcal{F}_t:\mathcal{A}_t\to\mathbb{R}\) 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 \(\sup_{x\in\mathbb{R}} |u_x^{(n)}(x,T)| \geq \alpha >0\), with \[ \lim_{n\to\infty} \left( \inf_{t\in [0,T]} \left( \sup_{x\in\mathbb{R}}|u_x^{(n)}(\cdot , t)| - \mathcal{F}_t (f_n , 0 , u^{(n)}) \right) \right) = 0 . \]