Corrigendum Non-trivial Non-negative Periodic Solutions of a System of Doubly Degenerate Parabolic Equations with Nonlocal Terms
چکیده
We correct a flaw in the proof of [1, Lemma 2.3]. 1. Corrigendum. This Corrigendum concerns the proof of [1, Lemma 2.3]. In that proof there is a flaw in the estimate of log xk due to an incorrect inequality. We provide here a correct estimate of log xk in 1.5 which preserves the validity of Lemma 2.3. For the reader convenience we recall the statement of Lemma 2.3 and we give its complete proof. Here m > 1 and p > 2. Lemma 2.3 Let K > 0 and assume that u is a non-negative periodic function such that u ∈ C(QT ), u ∈ L(0, T,W 1,p 0 (Ω)) and satisfying ut − div{[|∇(u + u)| + η] p−2 2 ∇(u + u)} ≤ Ku, in QT and u(·, t)|∂Ω = 0, for t ∈ [0, T ]. Then there exists R > 0 and independent of and η such that ‖u‖L∞ ≤ R. Proof. We follow Moser’s technique to show the stated a priori bounds. Multiplying ut − div{[|∇(u + u)| + η] p−2 2 ∇(u + u)} ≤ Ku by u, with s ≥ 0, integrating over Ω and passing to the limit as h → 0 in the Steklov averages uh we have K‖u(t)‖ Ls+2(Ω) ≥ 1 s+ 2 d dt ‖u(t)‖ Ls+2(Ω)