Gradient estimates for the heat equation in the exterior domains under the Neumann boundary condition. (English) Zbl 1240.35215

In this paper we study the decay rate of the gradient of the solution to the equation \(\partial _{t}u=\Delta u\) in \(\Omega \times (0,\infty )\) with Neumann boundary condition and the initial data \(\varphi \) without the convexity or the radial symmetry of the domain \(\Omega \). We prove the inequality \( \| (\nabla _{x}u)(t)\| _{L^{\infty }(\Omega )}\leq Ct^{-N/2p-1/2}\| \varphi \| _{L^{p}(\Omega )}\) for the case when \(N\geq 2\) and \(\Omega \) is the exterior smooth domain of a compact set in \(\mathbb {R}^N\).


35K05 Heat equation
35K20 Initial-boundary value problems for second-order parabolic equations
35B65 Smoothness and regularity of solutions to PDEs