\(L_ \infty\)-estimate for qualitatively bounded weak solutions of nonlinear degenerate diagonal parabolic systems (Q2785592)

We show only boundedness of qualitatively bounded weak solutions to the following Dirichlet problem for a diagonal parabolic system NEWLINE\[NEWLINEu_{it}-\text{div}(a_i(x,t,u,\nabla u)\cdot\nabla u_i)=b_i(x,t,u,\nabla u)\text{ in }\Omega^T,\quad u_i|_{t=0}=u_{0i}\text{ in }\Omega,\quad u_i=u_{bi}\text{ on }S^T,NEWLINE\]NEWLINE where \(i=1,\dots,m\), \(\Omega\subset\mathbb{R}^n\), \(\Omega^T=\Omega\times(0,T)\), \(S^T=S\times(0,T)\), \(S\) is the boundary of \(\Omega\), and dot denotes the scalar product in \(\mathbb{R}^n\). We assume the following growth conditions NEWLINE\[NEWLINEa_i(x,t,u,\nabla u)\cdot\nabla u_i\cdot \nabla u_i \geq\alpha_0|\nabla u|^{p-2}|\nabla u_i|^2-\varphi_{1i}(x,t),NEWLINE\]NEWLINE NEWLINE\[NEWLINEb_i(x,t,u,\nabla u)\leq\beta_0|\nabla u|^{p-2}|\nabla u_i|^2+\varphi_{2i}(x,t),NEWLINE\]NEWLINE where \(i=1,\dots,m\), \(\alpha_0\), \(\beta_0\) are positive constants, and \(\varphi_{1i}\), \(\varphi_{2i}\) are positive functions.