0
$\begingroup$

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.

Improve this question
$\endgroup$
5
  • $\begingroup$ What do you mean by "degenerate"? the coefficient of Laplacian vanishes somewhere on the manifold? If it does, then by using standard method, one can only derive interior estimate (for each $t$), namely, $L^p(\Omega)$ for $\Omega$ compactly contained in the regular portion. In particular, the bound will depend on the distance between degenerate points and the set $\Omega$. $\endgroup$
    – Chih-Wei Chen
    Commented Jul 8, 2014 at 16:55
  • $\begingroup$ Thanks for the comment. I was thinking of equations like porous medium equation by degenerate. $\endgroup$
    – TomJoseph
    Commented Jul 11, 2014 at 7:43
  • 5
    $\begingroup$ Questions like this always leave me with the impression that the poster has not done his/her homework to an appropriate extent. I mean, there are only about a bazillion or so papers in the literature about porous media and similar equations. So in a question like this, I would rather expect something along the lines of: I have read Refs. [1]-[20], and I have concluded X, but this still leaves Y unanswered. $\endgroup$
    – Michael Renardy
    Commented Sep 15, 2014 at 1:42
  • $\begingroup$ On a compact manifold without boundary, $u \equiv 1$ is a solution to the porous medium equation that lives in any $L^p$ space. So you cannot prove any estimates of the form you wrote with $f(t)$ decaying, at least without further assumptions on the allowed data. $\endgroup$
    – Willie Wong
    Commented Sep 16, 2014 at 15:00
  • 1
    $\begingroup$ @MichaelRenardy Did you miss that the estimate is desired on compact manifolds? On bounded domains yes of course there is plenty of work as I indicated in the OP; otherwise the quantity of literature is small. I am not sure how your somewhat rude comment gathered five upvotes. $\endgroup$
    – TomJoseph
    Commented Sep 22, 2014 at 14:11

1 Answer 1

Reset to default
0
$\begingroup$

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.

Improve this answer
$\endgroup$

Not the answer you're looking for? Browse other questions tagged .