Blow-up for semilinear parabolic equations in cones of the hyperbolic space

Abstract

Abstract We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type e μ t u p ( μ ∈ R , p > 1 ) , posed on cones of the hyperbolic space. Under a certain assumption on µ and p, related to the bottom of the spectrum of − Δ in H n , we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters µ and p satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters µ and p are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting (Bandle and Levine 1989 Trans. Am. Math. Soc. 316 595–622), and it is essentially due to a specific geometric feature of H n .