On Riemannian manifolds with negative sectional curvature, we study finite time blow-up and global existence of solutions to semilinear parabolic equations, where the power nonlinearity is multiplied by a time-dependent positive function h(t). 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, if h≡1 we have global existence for small initial data, whereas for h(t)=eαt a Fujita-type phenomenon appears for certain values of α>0. A key role will be played by the infimum of the L2-spectrum of the operator -δ on M. © 2011 Elsevier Inc.

Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature

PUNZO, FABIO
2012-01-01

Abstract

On Riemannian manifolds with negative sectional curvature, we study finite time blow-up and global existence of solutions to semilinear parabolic equations, where the power nonlinearity is multiplied by a time-dependent positive function h(t). 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, if h≡1 we have global existence for small initial data, whereas for h(t)=eαt a Fujita-type phenomenon appears for certain values of α>0. A key role will be played by the infimum of the L2-spectrum of the operator -δ on M. © 2011 Elsevier Inc.
Comparison principles; Finite time blow-up; Global existence; Ground states; Heat kernel; Laplace-Beltrami operator; Spectral analysis; Analysis; Applied Mathematics
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1028574
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 22
  • ???jsp.display-item.citation.isi??? 20
social impact