Dynamics of dissipative solutions to the Hardy-Sobolev parabolic equation

We study the long-time behaviour of solutions to the Hardy-Sobolev parabolic equation in critical function spaces for any spatial dimension $d\ge 5$. By employing the Fourier splitting method, we establish precise decay rates for dissipative solutions, meaning those whose critical norm vanishes as time approaches infinity. Our findings offer a deeper understanding of the asymptotic properties and dissipation mechanisms governing this equation.