$L^p$-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.