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

For radial initial data, we construct explicit higher-order $L^p({\mathbb{R}}^N)$-asymptotic profiles for the heat equation with Hardy potential. These profiles, denoted $A_n$ are obtained from the small-argument expansion, up to an arbitrary order $n$, of the modified Bessel function appearing in the radial Hardy heat kernel. If $u$ is the mild solution generated by this kernel, we prove that the corresponding remainder $u(x,t)-A_n(x,t)$ admits a polynomial decay depending on $n$ in $L^p({\mathbb{R}}^N)$ as $t\to \mathrm{\infty }$. We also treat the non-radial case through spherical harmonics: each angular mode evolves according to a radial Hardy heat equation with a modified parameter, leading to finite and infinite angular expansion versions of the asymptotic profile under suitable summability assumptions.