Classification of local asymptotics for solutions to heat equations with inverse-square potentials

TL;DR

This paper investigates the precise local asymptotic behavior of solutions to heat equations with inverse-square potentials, using advanced monotonicity and blow-up techniques, and establishes a unique continuation property.

Contribution

It introduces a novel combination of monotonicity formulas and blow-up methods to analyze solutions near singularities in heat equations with Hardy potentials.

Findings

Exact asymptotic behavior characterized near singularities

Unique continuation property established

Method applicable to linear and subcritical semilinear equations

Abstract

Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.