关键词: nonlinear parabolic equations, existence, Sobolev-Hardy inequality, singular potential

Abstract:

We study the existence of solutions to the following parabolic equation
{ut − Δ_pu =λ/|x|^s |u|^q−2 u,     (x, t) ∈ Ω × (0,∞),
u(x, 0) = f(x),                           x ∈Ω,
u(x, t) = 0,                               (x, t) ∈ ∂Ω × (0,∞),                   (P)
where −Δ_pu ≡ −div(|∇u|^p−2 ∇u), 1 < p < N, 0 < s ≤ p, p ≤ q ≤ p*(s) = N−s/N−p p, Ω is a bounded domain in RN such that 0 ∈ Ω with a C¹ boundary ∂Ω, f ≥ 0 satisfying some convenient regularity assumptions. The analysis reveals that the existence of solutions for (P) depends on p, q, s in general, and on the relation between  and the best constant in the Sobolev-Hardy inequality.

Key words: nonlinear parabolic equations, existence, Sobolev-Hardy inequality, singular potential