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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary
The paper Asymptotics of the porous media equation via Sobolev inequalities by Bonforte and Grillo seems to be what you require.
They key is the validity of a logarithmic Sobolev inequality.