Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations

Details Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations

2012-02-23

arxiv.org/abs/1202.5068v1

Mathematics

Analysis of PDEs

Scientific paper

We construct viscosity solutions to the nonlinear evolution equation \eqref{p} below which generalizes the motion of level sets by mean curvature (the latter corresponds to the case $p = 1$) using the regularization scheme as in \cite{ES1} and \cite{SZ}. The pointwise properties of such solutions, namely the comparison principles, convergence of solutions as $p\to 1$, large-time behavior and unweighted energy monotonicity are studied. We also prove a notable monotonicity formula for the weighted energy, thus generalizing Struwe's famous monotonicity formula for the heat equation ($p =2$).

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