Asymptotic profiles of solutions for a diffusion equation with a dynamic Hardy potential

We study the initial value problem for a linear diffusion equation with a time-dependent Hardy potential. It is shown that two types of solutions exist, each with different asymptotic profiles. Under an additional condition on the initial value, we derive the precise asymptotics of minimal positive solutions. To prove this, we construct appropriate comparison functions by modifying singular solutions of the linear heat equation. Using a similar approach, we investigate the existence of a larger solution and its asymptotic profile. These solutions are typical in the sense that any solution must have an asymptotic profile similar to either the minimal solution or the larger solution. To show this, we transform the equation into an integral equation and estimate the integral.