Ultracontractivity and convergence to equilibrium for supercritical quasilinear parabolic equations on Riemannian manifolds

Abstract

Let $(M,g)$ be a compact Riemannian manifold without boundary and dimension $d\ge3$. Let $u(t)$ be a solution to the problem $\dot u=\triangle_pu$, $u(0)=u_0$, $\triangle_p$ being the Riemannian $p$--Laplacian with $p>d$. Let also $\overline{u}$ be the (time--independent) mean of $u(t)$. We will prove ultracontractive estimates of the type $\Vert u(t)-\overline{u}\Vert_\infty\le C\Vert u(0)-\overline{u}\Vert_q^\gamma/t^\beta$. The constant $C$ depends only on $p$ and $q$, on geometric quantities of $M$ and on the dimension of the manifold, while the exponents $\beta$ and $\gamma$ depend only on $p$ and $q$ and differ according to the regimes $t\to0$ and $t\to+\infty$. Similar bounds hold when $\triangle_p$ is replaced by the subelliptic $p$--Laplacian associated to a collection of Hörmander vector fields. We also prove the L$^q$--L$^\infty$ Höolder continuity of the solutions, and apply similar methods to study the same questions for evolution equations on manifolds with boundary. The bounds are sharp in several of the above cases. The method relies on the theory of nonlinear Markov semigroups (\cite{CG2}) and on the connection between nonlinear ultracontractivity and logarithmic Sobolev inequality for the $p$--energy functional.