Sharp pointwise estimates for the gradients of solutions to linear parabolic second order equation in the layer

TL;DR

This paper derives explicit sharp pointwise gradient estimates for solutions to linear parabolic equations with constant coefficients, considering both homogeneous and nonhomogeneous cases in a bounded time layer.

Contribution

It provides explicit formulas for the sharp coefficients in pointwise gradient estimates for solutions to linear parabolic equations, extending understanding of gradient behavior in these problems.

Findings

Explicit formulas for sharp gradient estimate coefficients

Pointwise gradient bounds for solutions with initial data in L^p spaces

Gradient estimates for nonhomogeneous equations with right-hand side in L^p and Hölder spaces

Abstract

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second order equations with real constant coefficients in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$ and $T<\infty$. The homogeneous equation is considered with initial data in $L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^\alpha \big (\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $\alpha \in (0, 1)$. Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained.