Fast diffusion equations on Riemannian manifolds

Abstract

In the present paper, we first study the nonexistence of positive solutions of the following nonlinear parabolic problem \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\Delta_g( u^m)+V(x)u^m+\lambda u^q & \text{in}\quad \Omega \times (0, T ), \\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T). \end{cases} \end{equation*} Here, $\Omega$ is a bounded domain with smooth boundary in a complete non-compact Riemannian manifold $M$, $0 < m < 1$, $V\in L_{\text{loc}}^1(\Omega)$, $ q > 0 $ and $\lambda\in \mathbb{R}$. Next, we prove some Hardy and Leray type inequalities with remainders on a Riemannian Manifold $M$. Furthermore, we obtain explicit (sometimes optimal) constants for these inequalities and present several nonexistence results with help of Hardy and Leray type inequalities on the hyperbolic space $\mathbb{H}^n$.