The self-similar solutions to a fast diffusion equation
The author considers the equation \[ \Delta u^\alpha - {x \cdot \nabla u \over 2} - \mu u + u^\beta = 0 \text{ in } \mathbb{R}^n, \tag{1} \] where \(0 < \alpha < 1\), \(\beta > 1\) and \(\mu > 0\) are parameters. This equation arises from the study of blow up of self-similar solutions of the fast diffusion equation (2) \(\psi_t = \Delta \psi^\alpha + \psi^\beta\). Considering radial solutions of (1) \(u(x) = u(r)\), where \(r=|x |\) and setting \(u^\alpha (r) = w (\alpha^{-m} r) \alpha^{- \alpha /(\beta - 1)}\) we can receive an equation for \(w\): \[ w'' + {n - 1 \over r} w' - \left( {r \over 2} w' + \lambda w \right) w^q + w^p = 0, \quad w'(0) = 0,\;w(0) = \eta > 0, \tag{3} \] where \(p = \beta/ \alpha\), \(q = (1 - \alpha)/ \alpha\) and \(\lambda = \alpha \mu\). In this paper conditions on \(p,q,n\) and \(\lambda\) which imply the existence or non-existence of ground state of the equation (3) are established. Some other results on the existence and nonexistence of self-similar solutions of the equation (2) and on characterization the local behaviour of solutions of equation (2) near a blow-up point are derived.
scientific article; zbMATH DE number 754595
scientific article