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

We show a global existence result for a doubly nonlinear porous medium type equation of the form ut=Δpum+uq on a complete and non-compact Riemannian manifold M of infinite volume. Here, for 11 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 [Formula presented] and the initial datum is small enough; namely, we establish an explicit bound on the L∞ norm of the solution at all positive times, in terms of the L1 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.