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

Abstract

The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation ut- Δu - V(x)u = 0 in ℝnwith singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely, PMk-spaces. For arbitrary data u0∈ PMk, 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 [Formula: see text] with [Formula: see text]. This threshold value is the same one obtained for the global well-posedness of L2-solutions by means of Hardy inequalities and energy estimates. Since there is no any inclusion relation between L2and PMk, our results indicate that λ*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 PMk.