Radial fast diffusion on the hyperbolic space (Q2921103)

The authors study the asymptotic behavior of positive radial solutions to the fast diffusion equation on \(N\)-dimensional hyperbolic space \(H^N\) space for \(N\geq 2\): NEWLINE\[NEWLINE u_t = \Delta(u^m), NEWLINE\]NEWLINE where \(\Delta\) is the Riemannian Laplacian, \(m \in (m_s,1)\) and \(m_s\) is the critical exponent, \(m_s = (N-2)/(N+2)\). Fine asymptotics are studied near the extinction time \(T\) in terms of a separable solution of the form \(\nu(r,t) = (1 - t/T)^{1/(1 - m)}V^{1/m}(r)\), where \(V\) is the unique positive energy solution, radial with respect to \(0\), to NEWLINE\[NEWLINE -\Delta V = cV^{1/m} NEWLINE\]NEWLINE for a suitable \(c > 0\).NEWLINENEWLINEAs the main result, the authors show that \(u\) converges to \(\nu\) in relative error i.e. \(\|u^m/\nu^m(\cdot,t) - 1\|_{\infty} \to 0\), as \(t \to T^-\). In particular, the solution is bounded above and below, near the extinction time \(T\), by multiples of \((1 - t/T)^{1/(1-m)}e^{-(N-1)r/m}\). In particular, sharp convergence results as \(t \to T^-\) are shown for spatial derivatives in the form of convergence in relative error.