Semilinear heat equation with singular terms - Ould Khatri Mohamed Mahmoud; Youssfi Ahmed (2022)
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 ∂tu − ∆u = µ u |x| 2 f u in ΩT := Ω × (0, T), u = 0 on ∂Ω × (0, T), u(x, 0) = u0(x) in Ω, where Ω ⊂ RN, N ≥ 3, is a bounded open, σ ≥ 0 and µ > 0 are real constants and f ∈ L m(ΩT), m ≥ 1, and u0 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.
Keywords: Természettudományok; Matematika