  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.
