Existence of mild solutions for a singular parabolic equation and stabilization

Abstract

In this paper, we study the existence and the uniqueness of a positive mild solution for the following singular nonlinear problem with homogeneous Dirichlet boundary conditions: (St) ∂tu - Δpu = u -δ + f(x,u) in (0,T) × Ω =: QT, u = 0 on (0,T) × ∂Ω, u > 0 in QT, u(0,x) = u0 ≥ 0 in Ω, where Ω stands for a regular bounded domain of ℝN, Δpu is the p-Laplacian operator defined by Δpu = div(|∇u|p-2|∇u|) 1 0 and T > 0. The nonlinear term f : Ω × ℝ → ℝ is a bounded below Carathéodory function and nonincreasing with respect to the second variable (for a.e. x ∈ Ω). We prove for any initial positive data u0 ∈ 𝒟(A)¯L∞$\overline{{\mathcal {D}}(A)}^{L^\infty }$ the existence of a mild solution to (St). Then, we deduce some stabilization results for problem (St) in L∞(Ω) when p ≥ 2. This complements some results obtained in [J. Differential Equations 252 (2012), 5042–5075] stated with the additional restriction δ < 2 + 1/(p - 1).

Keywords

• singular parabolic equation
• p-laplace equation
• nonlinear semi-group theory
• 35k67
• 35k92
• 58d07