$L^p$-Asymptotic Profiles for the Heat Equation with a Hardy Potential

For radial initial data, we construct explicit higher-order asymptotic profiles in $L^p(\mathbb{R}^N)$ to the heat equation with inverse-square potential.

These profiles are obtained from the small-argument expansion (up to arbitrary order $n$) of the modified Bessel function appearing in the radial heat kernel.

We prove that, after subtracting the profile of order \(n\), the remainder decays faster by an additional power of \(t\).

We also describe the non-radial case through spherical harmonics: each angular mode evolves according to a radial Hardy heat equation, leading to finite and infinite angular expansion versions of the asymptotic profile under suitable summability assumptions.