Abstract
The main goal of this paper is to analyze the existence and nonexistence as well as the regularity of positive solutions for the following initial parabolic problem { ∂ t u − Δ u = μ u | x | 2 + f u σ in Ω T := Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where Ω ⊂ R N , N ≥ 3 , is a bounded open, σ ≥ 0 and μ > 0 are real constants and f ∈ L m ( Ω T ) , m ≥ 1 , and u 0 are nonnegative functions. The study we lead shows that the existence of solutions depends on σ and the summability of the datum f as well as on the interplay between μ and the best constant in the Hardy inequality. Regularity results of solutions, when they exist, are also provided. Furthermore, we prove uniqueness of finite energy solutions.
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