Global solutions of semilinear parabolic equations with drift term on Riemannian manifolds

Abstract

<p style='text-indent:20px;'>We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term <inline-formula><tex-math id="M1">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any <inline-formula><tex-math id="M2">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula>, depending on the vector field <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula>, global solutions exist, for sufficiently small initial data.</p>