Global existence for a Leibenson type equation with reaction on Riemannian manifolds

Abstract

We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t={\Delta }_p{u}^m+u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume $m(p-1)\ge 1$, $m>1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac{p}{N}$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^{\infty }$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure.