Remarks on global bounds of solutions of parabolic equations in divergence form (Q1122703)

Consider the Dirichlet problems of parabolic equations \[ (1)\quad u_ t=\sum a_ i(t,x,u,u_ x)_{x_ i}+a(t,x,u,u_ x)u+f(t,x);\quad u=0\quad on\quad \partial \Omega \subset R^ n \] and \[ (2)\quad u_ t=u_{xx}+f(t,x,u,u_ x,u_{xx},u_ t);\quad u_ x=0\quad on\quad \partial \Omega. \] In this paper some properties of solutions of (1) are proved with the proof of a variant of the maximum principle and global boundedness of the spatial derivative \(u_ x\) for (2) with bounded perturbation \(f(t,x,u,u_ x,u_{xx},u_ t)\) are studied.