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\(L_ \infty\)-estimate for qualitatively bounded weak solutions of nonlinear degenerate diagonal parabolic systems. (English) Zbl 0867.35055

We show only boundedness of qualitatively bounded weak solutions to the following Dirichlet problem for a diagonal parabolic system \[ u_{it}-\text{div}(a_i(x,t,u,\nabla u)\cdot\nabla u_i)=b_i(x,t,u,\nabla u)\text{ in }\Omega^T,\quad u_i|_{t=0}=u_{0i}\text{ in }\Omega,\quad u_i=u_{bi}\text{ on }S^T, \] 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 \[ a_i(x,t,u,\nabla u)\cdot\nabla u_i\cdot \nabla u_i \geq\alpha_0|\nabla u|^{p-2}|\nabla u_i|^2-\varphi_{1i}(x,t), \]
\[ b_i(x,t,u,\nabla u)\leq\beta_0|\nabla u|^{p-2}|\nabla u_i|^2+\varphi_{2i}(x,t), \] where \(i=1,\dots,m\), \(\alpha_0\), \(\beta_0\) are positive constants, and \(\varphi_{1i}\), \(\varphi_{2i}\) are positive functions.

MSC:

35K65 Degenerate parabolic equations
35B50 Maximum principles in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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