Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

TL;DR

This paper investigates the conditions under which solutions to a semi-linear heat equation on hyperbolic space either exist globally or blow up in finite time, focusing on different nonlinearities and weights, and extends results to Cartan-Hadamard manifolds.

Contribution

It identifies nonlinearities that induce Fujita phenomena for power weights and analyzes exponential nonlinearities, extending results to hyperbolic space and Cartan-Hadamard manifolds.

Findings

Existence of critical exponents for blow-up versus global solutions.

Fujita phenomena occur for exponential weights with a critical parameter.

Extension of results to hyperbolic and Cartan-Hadamard manifolds.

Abstract

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space: \begin{align}\label{abs:eqn} \left\{\begin{array}{ll} \partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) &\hbox{ in }~ \mathbb{H}^{n}\times (0, T),\\ \\ \quad u =u_{0} &\hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, it exhibits Fujita phenomena for the exponential weight $h(t) = e^{\mu t},$ i.e. there exists a critical exponent $\mu^*$ such that if $\mu > \mu^*$ then all non-negative solutions blow-up in finite time and if $\mu \leq \mu^*$ there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that the above mentioned Cauchy problem with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.