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Degenerate singular parabolic problems with natural growth

El Ouardy Mounim, El Hadfi Youssef, Sbai Abdelaaziz

Opuscula Mathematica

2024

Vol. 44, no. 4

471--503

In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term [formula] where Ω is a bounded open subset of RN, N > 2, Q is the cylinder Ω × (0, T), T > 0, Γ the lateral surface ∂Ω×(0, T), 2 ≤ p < N, a(x, t) and b(x, t) are positive measurable bounded functions, q ≥ 0, 0 ≤ γ < 1, and ƒ non-negative function belongs to the Lebesgue space Lm(Q) with m > 1, and u0 ∈ L∞(Ω) such that ∀ω ⊂⊂ Ω∃Dω > 0 : u0 ≥ Dω in ω. More precisely, we study the interaction between the term uq (q > 0) and the singular lower order term d(x, t)|∇u|pu−γ (0 < γ < 1) in order to get a solution to the above problem. The regularizing effect of the term uq on the regularity of the solution and its gradient is also analyzed.