Parabolic p-systems: boundary regularity

We will establish everywhere regularity up the boundary for weak solutions of the parabolic system 1.1$$ \left\{ {\begin{array}{*{20}{c}} {u \equiv ({u_1},{u_2}, \ldots,{u_m}), m \in N,} \\ {{u_i} \in C(\varepsilon,T,{L^2}(\Omega )) \cap {L^p}(\varepsilon,T,{W^{1,p}}(\Omega )),} \\ {{u_{i,t}} - div{{\left| {Du} \right|}^{p - 2}}D{u_i} = {B_i}(x,t,u,Du),in \Omega \times (\varepsilon,T)} \\ {\varepsilon \in (0,T),i = 1,2, \ldots,m,p > \max \left\{ {1;\frac{{2N}}{{N + 2}}} \right\},} \end{array}} \right. $$ associated with Dirichlet boundary data 1.2$$ {u_i}( \cdot,t) = {g_i}( \cdot,t) on \partial \Omega \times(\varepsilon,T), $$ in the sense of the traces on, of functions in W1P(,f2). The basic assumptions on 8.f2, the boundary data g and the forcing term B$$ g \equiv ({g_1},{g_2}, \ldots,{g_m}), B \equiv ({B_1},{B_2}, \ldots,{B_m}), $$ are the following: ∂Ωis of classC1,λ;for some λ ∈(0,1), in the sense of (1.2) of Chap. I. Thus the norm $$ {\left| {\left\| {\partial \Omega } \right\|} \right|_{1 + \lambda }} $$is finite. $${g_i},i = 1,2, \ldots,m,$$are restrictions to ∂Ω of functions ğ _i defined in the whole Ω _T , and satisfying is of class for some in the sense of (1.2) of Chap. I. Thus the norm is finite.The functions, are restrictions to of functions defined in the whole, and satisfying