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

TL;DR

This paper studies when solutions to a semilinear heat equation on hyperbolic cones blow up or exist globally, revealing that geometry significantly influences solution behavior unlike in Euclidean spaces.

Contribution

It establishes optimal conditions on parameters for blow-up or global existence of solutions on hyperbolic cones, highlighting geometric effects.

Findings

Solutions blow up in finite time under certain parameter conditions.

Existence of global solutions depends on initial data size when parameters are opposite.

Blow-up and global existence are independent of cone amplitude, unlike Euclidean cases.

Abstract

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{\mu t}u^p$ ($\mu\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $\mu$ and $p$, related to the bottom of the spectrum of $-\Delta$ in $\mathbb H^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $\mu$ 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 $\mu$ 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 $\mathbb H^n$.