Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds (Q1710740)

The authors consider the nonlinear evolution problem for the porous medium equation \[ \begin{cases} u_t=\Delta(u^m)+u^p & \text{in}\;M\times (0,T),\\ u=u_0 & \text{in}\;M\times \{0\}, \end{cases} \] in the slow diffusion case \(m>1,\) where \(M\) is an \(N\)-dimensional complete, simply connected Riemannian manifold with nonpositive sectional curvatures, that is, a Cartan-Hadamard manifold, \(\Delta\) is the Laplace-Beltrami operator on \(M,\) \(p>1,\) \(T\in(0,\infty],\) and the initial data \(u_0\) is nonnegative, bounded and compactly supported. The main results obtained can be summarized as follows. If \(p>m,\) then global-in-time solutions exist for small data \(u_0,\) while solutions associated to large data blow up in finite time. In the case \(p < m,\) all sufficiently large data give rise to solutions blowing up at worst in infinite time. If the stronger restriction \(p\in \big(1,(1+m)/2\big]\) holds, then all data give rise to solutions existing globally in time.