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Gradient estimates for Dirichlet parabolic problems in unbounded domains. (English) Zbl 1061.35022
The authors consider the following Dirichlet problem: \[ \begin{cases} u_t(t,x)-Au(t,x)=0, & t\in(0,T),\;x\in \Omega,\\ u(t,\xi)=0, & t\in(0,T),\;\xi\in \partial\Omega,\\ u(0,x)=f(x), & x\in \Omega, \end{cases} \] where \(f\) is continuous and bounded in an unbounded smooth connected open set \(\Omega\subset {\mathbb R}^N.\) The operator \(A\) is a second-order elliptic one, with (possibly) unbounded regular coefficients. The authors determine new conditions on the coefficients of \(A\) yielding global estimates for the bounded classical solution.

MSC:
35B45 A priori estimates in context of PDEs
35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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