Boundary regularity for parabolic systems with nonstandard (p,q)-growth conditions in smooth convex domains

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form aligned ∂ ₜ uⁱ - div ( a(|Du|) Duⁱ )= fⁱ, i=1, ,N, aligned in a space-time cylinder Ω T = Ω × (0,T) , where Ω ⊂ Rⁿ ( n ≥ 2 ) is a bounded, convex C² -domain and $$T>0$$ . The inhomogeneity f=(f¹, ,fN) belongs to Ln+2+σ(Ω T,RN) for some σ >0 . The coefficients a:R>0 → R>0 are of Uhlenbeck-type and satisfy a nonstandard ( p , q )-growth condition with 2 ≤ p ≤ q < p + 4/n+2. Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.