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A gradient estimate for solutions of the heat equation. (English) Zbl 0951.35017
It is shown that the solution \(u(x,t)\) of the higher dimensional heat equation \[ u_t = \Delta _x u \;\text{in} \;\Omega \times (0,\infty), \;u= 0 \;\text{on} \;\partial \Omega \times (0,\infty), \;u(x,0) = f(x), \;x\in \Omega , \] with \(\Omega \subset \mathbb{R}^n\) bounded, \(n\geq 2\), satisfies \[ |\nabla _x u(x,t)|\leq \max _{\overline {\Omega }} |\nabla f|\tag \(*\) \] provided that \(f\in C^1\), \(f=0\) on \(\partial \Omega \), \(\partial \Omega \) is \(C^3\) and satisfies a certain mean curvature condition. An example shows that the estimate \((*)\) fails if the last condition does not hold.
Reviewer: A.Kufner (Praha)

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
35K20 Initial-boundary value problems for second-order parabolic equations
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References:
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[2] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, 1968.
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