Some remarks on quasilinear parabolic problems with singular potential and a reaction term
The authors consider the following quasilinear parabolic equation \[ (u^{\theta})_t - \Delta_pu = \lambda\frac{u^{p-1}|}{|x|^p} + u^q + f(x,t), \] where \(\theta\) is either 1 or \(p-1\), \(N\geq3\), \(\Omega\) is either a bounded regular domain containing the origin or \(\Omega\) is equal to all the space, \(1<p<N\), \(q>0\) and \(f\geq 0\) with suitable hypotheses. Such kind of problems are related to the classical Hardy inequality. The authors obtain natural conditions to show the existence or the nonexistence of nonnegative solutions. In the case of nonexistence result, the authors analyse blow-up phenomena for approximated problems in connection with a strong maximum principle and with the Harnack inequality. They also study when finite time extinction (\(1<p<2\)) and finite speed propagation (\(p>2\)) occur giving sharp conditions on the reaction power.
scientific article; zbMATH DE number 6340663
scientific article