I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|\_{L^p} \leq f(t)|u\_0|\_{L^q}$$
for the usual type of functions $f$.


The problem I have with adapting the techniques from the standard Euclidean case (eg. bounded domain) is the lack of Poincare's inequality and Nash inequality in the convenient form. So I guess there is a different way to do this. So please do recommend me a source where this is done for compact manifolds. Thank you.



Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary
The paper *[Asymptotics of the porous media equation via Sobolev inequalities](http://www.sciencedirect.com/science/article/pii/S0022123605001291)* by Bonforte and Grillo seems to be what you require.


They key is the validity of a logarithmic Sobolev inequality.


