On the Schr\"odinger-Debye System in Compact Riemannian Manifolds

We consider the initial value problem (IVP) associated to the Schr\"odinger-Debye system posed on a $d$-dimensional compact Riemannian manifold $M$ and prove local well-posedness result for given data $(u_0, v_0)\in H^s(M)\times (H^s(M)\cap L^{\infty}(M))$ whenever $s>\frac{d}2-\frac12$, $d\geq 2$. For $d=2$, we apply a sharp version of the Gagliardo-Nirenberg inequality in compact manifold to derive an a priori estimate for the $H^1$-solution and use it to prove the global well-posedness result in this space.