Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.

For any positive function $v$, I set

$$ J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g. $$

Assume now that $(v_t)_{t \geq 0}$ is a solution of the heat equation $$ \frac{d}{dt} v_t = \Delta_g v_t. $$

Is it possible to prove the following statement?

There exists a constant $\lambda$ independent of the initial data $v_0$ such that $$ J(v_t) \leq J(v_0) e^{\lambda t} $$

Computing the time derivative of $J(v_t)$ does not seem to be the solution since the expression we obtain is quite intricate (the initial problem is to prove that this derivative is "controllable" in some sense)

Any help would be invaluable!

As a reward: This question appeared in the course of proving a gradient estimate for the so called Lichnerowicz equation (see e.g. https://arxiv.org/abs/1403.5655). I will be happy to add the name of the first who finds the solution to the list of authors.

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