September/October 2023 Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up
Debdip Ganguly, Debabrata Karmakar, Saikat Mazumdar
Adv. Differential Equations 28(9/10): 779-805 (September/October 2023). DOI: 10.57262/ade028-0910-779


We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space : \begin{align} \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.\tag{0.1}\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, (0.1) exhibitsFujita 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(0.1)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.


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Debdip Ganguly. Debabrata Karmakar. Saikat Mazumdar. "Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up." Adv. Differential Equations 28 (9/10) 779 - 805, September/October 2023.


Published: September/October 2023
First available in Project Euclid: 25 May 2023

Digital Object Identifier: 10.57262/ade028-0910-779

Primary: 35A01 , 35R01 , 58J35

Rights: Copyright © 2023 Khayyam Publishing, Inc.


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Vol.28 • No. 9/10 • September/October 2023
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