The aim of this paper is to investigate the well-posedness of the Cauchy problem ∂_t u=∆u +V(x) u in R^N × (0, T ), N ≥ 3 u(x,0)=u_0(x) on R^N where the potential V is defined by V=V(x) := λ/|x|^2 , 0≤λ<(N − 2)^2 /4. Roughly speaking, we prove that a sufficient condition for existence and uniqueness of the solution is to restrict the growths of the solution u and of the initial datum u_0 as |x| → ∞ (at most like e^{c|x|^2}, with c ∈ R+) and near the origin (at most like k|x|^α , with k ∈ R_+ , while α is a parameter depending on λ). For λ > 0, the solution shall present a lack of regularity in the origin which is due only to the presence of the singular potential.
The Cauchy problem for the Heat Equation with a Singular Potential
MARCHI, CLAUDIO
2003
Abstract
The aim of this paper is to investigate the well-posedness of the Cauchy problem ∂_t u=∆u +V(x) u in R^N × (0, T ), N ≥ 3 u(x,0)=u_0(x) on R^N where the potential V is defined by V=V(x) := λ/|x|^2 , 0≤λ<(N − 2)^2 /4. Roughly speaking, we prove that a sufficient condition for existence and uniqueness of the solution is to restrict the growths of the solution u and of the initial datum u_0 as |x| → ∞ (at most like e^{c|x|^2}, with c ∈ R+) and near the origin (at most like k|x|^α , with k ∈ R_+ , while α is a parameter depending on λ). For λ > 0, the solution shall present a lack of regularity in the origin which is due only to the presence of the singular potential.Pubblicazioni consigliate
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