Universal estimate of the gradient for parabolic equations

TL;DR

This paper introduces a universal estimate for the gradient of solutions to parabolic equations, incorporating higher order coefficients into the weight, with applications to asymptotic behavior and regularity results.

Contribution

It proposes a modified upper limit estimate for weighted Sobolev norms of parabolic solutions that includes higher order coefficients in the weight, applicable across various settings.

Findings

Derived a universal gradient estimate independent of dimension and coefficients

Obtained asymptotic estimates for the gradient at initial time

Established existence and regularity results for delayed parabolic equations

Abstract

We suggest a modification of the estimate for weighted Sobolev norms of solutions of parabolic equations such that the matrix of the higher order coefficients is included into the weight for the gradient. More precisely, we found the upper limit estimate that can be achieved by variations of the zero order coefficient. As an example of applications, an asymptotic estimate was obtained for the gradient at initial time. The constant in the estimates is the same for all possible choices of the dimension, domain, time horizon, and the coefficients of the parabolic equation. As an another example of application, existence and regularity results are obtained for parabolic equations with time delay for the gradient.