Total blow-up versus single point blow-up
The authors study blow-up of non negative solutions to a quasilinear parabolic equation describing the temperature perturbation during the ignition process of a reactive gas in a spherical container. The equation is of the form \(u_ t-\Delta u=f(u)+g(t),\) and is supplemented with zero Dirichlet boundary conditions and with a non negative initial datum which is radially symmetric and radially decreasing. The authors assume f to be differentiable and non negative for non negative arguments; g is either a non negative given function of time, or K times the space average of \(u_ t\), with \(0<K<1\). The latter assumption for g should model the compressibility of the gas. The authors provide several sufficient conditions in order blow-up to occur only at the origin, respectively everywhere. These conditions involve explicit assumptions for the function f and, in some case, on the smallness of the parameter K.
scientific article; zbMATH DE number 4103597
scientific article