Semilinear limit problems for reaction–diffusion equations with variable exponents

Convergence of quasilinear parabolic equations to semilinear equations

“…Introduction. In this paper we improve the results presented in our previous work [3]. We proved in [3] a weak upper semicontinuity of the global attractors for problem (2) as j → +∞.…”

“…In this paper we improve the results presented in our previous work [3]. We proved in [3] a weak upper semicontinuity of the global attractors for problem (2) as j → +∞. In this work we will prove that in fact we can obtain more, i.e., we can obtain upper semicontinuity of the global attractors, but for this, we have to revisit the uniform estimates of the solutions and to prove the continuity of the flow in another way.…”

“…Final remarks. In the way that we proved the continuity of the flow in the previous work [3] it was necessary to use |∇u(x, t)| + |u(x, t)| ≤ C, for a.e. x ∈ Ω,…”

“…Let {S(t)u 0 := u(t, u 0 )} t≥0 be the unique mild solution of (4) in L 2 (Ω) (see [3] for details).…”

“…for any t ∈ [0, T ], for some constant C = C(T ) > 0 (see page 3917 in [3]). So, we have to express our thanks to Yu Yangyang and Rong-Nian Wang for pointing out a problem in the proof of Theorem 3 in [3] and here we give a corrigendum to Theorem 3 in [3], because it is only true for N = 1 : Since H α → L ∞ (Ω) for α > N…”