Quasi-linear parabolic systems in divergence form with weak monotonicity (Q1847827)

The initial-boundary value problem for the parabolic system \[ u_t - \operatorname {div}a(x,t,u,Du)=f \] is considered. Solvability results in the natural energetic space \(L_p (0,T; W_0^{1,p}(Q))\) are established under standard continuity, growth and coercivity conditions on \(a\), but only mild monotonicity assumptions. One of the monotonicity assumptions which is proved to be sufficient for existence of a weak solution is the strict \(p\)-quasi-monotonicity of \(a\) in the last argument; another is that \(a\) is a gradient in the last argument of the convex and \(C^1\) function.