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

2022/06/23 by Monticelli, Dario D., Punzo, Fabio
#35A01 #35B44 #35K91 #58J35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.11758

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^μtu^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 \mathbb Hⁿ, 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, and it is essentially due to a specific geometric feature of \mathbb Hⁿ.

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