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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:
 [1] A. Friedman: Partial Differential Equations of Parabolic Type. Prentice Hall Inc., Englewood Cliffs, 1964. · Zbl 0144.34903 [2] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, 1968. [3] M. H. Protter and H. F. Weinberger: Maximum Principles in Differential Equations. Prentice Hall Inc., Englewood Cliffs, 1967. · Zbl 0153.13602
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