Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Fabio Punzo [1]

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

Pages: 113-138
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.
[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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