GLOBAL SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS WITH DRIFT TERM ON RIEMANNIAN MANIFOLDS

We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term u(P), on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any p > 1, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any p > 1, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain p depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of p, depending on the vector field b, global solutions exist, for sufficiently small initial data.