**Fabio Punzo** ^{[1]}

*
Global existence of solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature
*

**Pages:** 113-138

**Received:** 30 April 2013

**Accepted in revised form:** 30 July 2013

**Mathematics Subject Classification (2010):** 35B51, 35B44, 35K08, 35K58, 35R01.

**Keywords:** Local existence, finite time blow-up, global existence, mild solutions, Laplace-Beltrami operator, heat
kernel.

**Author addresse:**

[1] : Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy.

**Abstract:**
We address local existence,
blow-up and global existence of mild solutions to the semilinear
heat equation on Riemannian manifolds with negative sectional
curvature. We deal with a power nonlinearity multiplied by a
time-dependent positive function \(h(t)\), and initial conditions
\(u_0\in L^p(M)\). We show that depending on the behavior at
infinity of \(h\), either every solution blows up in finite time, or
a global solution exists, if the initial datum is small enough. In
particular, for any power nonlinearity, if \(h\equiv 1\) we have
global existence for small initial data, whereas if \(h(t)=e^{\alpha
t}\) a Fujita type phenomenon prevails varying the parameter
\(\alpha>0\).

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