Lower Estimates of L∞-norm of Gradients for Cauchy Problems

Journal of mathematical analysis and applications（2018）

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摘要

In this paper, we consider the Cauchy problems for a uniformly parabolic equation and a Hamilton–Jacobi equation. Our aim is to show the optimality of an upper estimate of the L∞-norm of the gradient of solution to each Cauchy problem. In order to achieve this aim, we investigate a lower estimate of this L∞-norm.

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关键词

Lipschitz constants,Lower estimates,Optimality,Parabolic equations,Hamilton Jacobi equations,Logarithmic Sobolev inequality

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