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

2013/07/24 by Ferreira, Lucas C. F., Mesquita, Cláudia Aline A. S.
#35A01 #35B06 #35B09 #35C06 #35K05 #35K67 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.6464

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 u_t-Δu-V(x)u=0 in ℝⁿ with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely PM^k-spaces. For arbitrary data u₀∈ PM^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)=\fracλ|x|² with |λ|

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