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

Abstract

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 ${\sup }_{x\in \mathbb{R}}|u_x^{(n)}(x,T)|\ge \alpha >0$, with \[ \lim_{n\to\infty} \left( \inf_{t\in [0,T]} \left( \sup_{x\in\mathbb{R}}|u_x^{(n)}(\cdot , t)| - \mathcal{F}_t (f_n , 0 , u^{(n)}) \right) \right) = 0 . \]

arXiv comment18 pages

References in corpus (1)

• The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data

Cited by in corpus (1)

• On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity