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Gradient estimates for some quasilinear parabolic equations with nonmonotonic perturbations. (English) Zbl 0865.35018

The author considers the following initial boundary value problem for the quasilinear parabolic equation \(u_t-\text{div}(|\nabla u|^m\nabla u)+g(x,u)=0\) in \(\Omega\times(0,\infty)\), \(u(x,0)=u_0(x)\), and \(u(x,t)|_{\partial\Omega}=0\) for \(t\in \mathbb{R}^+\equiv[0,\infty)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with a smooth boundary, \(m\) is a positive constant and \(g\) is a continuously differentiable function on \(\overline\Omega\times\mathbb{R}^+\) satisfying the estimate \(|g(x,u)|\leq k|u|^{\alpha+1}\) with some \(\alpha>m\) and \(k>0\). He derives an \(L^\infty\)-estimate for \(|\nabla u(t)|\) under the assumption \(u_0\in L^p\), \(p\geq 2\).

MSC:

35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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