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