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

作者：Ferreira Lucas C F^*; Mesquita Claudia Aline A S

来源：Communications in Contemporary Mathematics, 2015, 17(5): 1550041.

DOI：10.1142/S0219199715500418

• Linear heat equation
• singular potentials
• global existence
• asymptotic behavior
• PMk-spaces

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) - Delta u - V (x)u = 0 in R-n with 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 u(0) is an element of 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 V (x) = lambda/vertical bar x vertical bar(2) with vertical bar lambda vertical bar < lambda(*) = (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 PMk, our results indicate that lambda(*) 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.

• 出版日期2015-10