Semilinear heat equation with singular terms

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.