Semilinear heat equation with singular terms

The main goal of this paper is to analyze the existence and nonexistence as well regularity positive solutions for following initial parabolic problem { ∂ t u − − --> mathvariant="normal">Δ Δ = μ μ | x 2 + f σ σ displaystyle="false" in mathvariant="normal">Ω Ω T := × × stretchy="false">( 0 , stretchy="false">) on fence="true" stretchy="true" symmetric="true" /> where xmlns="http://www.w3.org/1998/Math/MathML"> ⊂ ⊂ mathvariant="double-struck">R N , ≥ ≥ 3 a bounded open, > are real constants ∈ ∈ L m 1 nonnegative functions. study we lead shows that depends on summability datum interplay between best constant in Hardy inequality. Regularity results solutions, when they exist, also provided. Furthermore, prove uniqueness finite energy solutions.