An approach without using Hardy inequality for the linear heat equation with singular potential

The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation $u_t-\mathrm{\Delta }u-V(x)u=0$ in ${\mathbb{R}}^n$ with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely $P{M}^k$-spaces. For arbitrary data $u_0\in P{M}^k$, the approach allows to compute an explicit smallness condition on $V$ for global existence in the case of $V$ with finitely many inverse square singularities. As a consequence, well-posedness of solutions is obtained for the case of the monopolar potential $V(x)=\frac{\lambda }{{\mid x\mid }^2}$ with $\mid \lambda \mid <{\lambda }_{\ast }=\frac{(n-2{)}^2}{4}$. This threshold value is the same one obtained for the global well-posedness of $L^2$-solutions by means of Hardy inequalities and energy estimates. Since there is no any inclusion relation between $L^2$ and $P{M}^k$, our results indicate that ${\lambda }_{\ast }$ is intrinsic of the PDE and independent of a particular approach. We also analyze the long time behavior of solutions and show there are infinitely many possible asymptotics characterized by the cells of a disjoint partition of the initial data class $P{M}^k$.