Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

Abstract

We consider the homogeneous Dirichlet problem for the parabolic equation u t - div (|∇u|p(x, t)-2 ∇u) = f (x, t) + F (x, t, u, ∇u) in the cylinder QT:= ω × (0, T):=, where ω ⊂ ℝN, N ≥ 2, is a C2-smooth or convex bounded domain. It is assumed that p ϵ C0,1 (Q ¯T) is a given function and that the nonlinear source F (x, t, s, ζ) has a proper power growth with respect to s and ζ. It is shown that if p (x, t) > 2(N + 1)/N + 2, f ϵ L2 (QT), |∇u0|p (x, 0) ϵ L1 (ω), then the problem has a solution u ϵ C0 ([ 0, T ]; L 2 (ω)) with |∇u|p (x, t) ϵ L∞ (0, T; L1 (ω)), ut ϵ L2 (QT), obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: |∇ u|2(p (x, t) - 1) + r ϵ L1 (QT), for any 0 < r < 4/N + 2, |∇ u |p (x, t) 2 ∇u ϵ L2 (0, T; W 1, 2 (ω))N. © 2024 the author(s), published by De Gruyter.