Blow-up phenomena for polytropic equation with inhomogeneous density and source

The subject of this investigation is the blow-up phenomena of the positive solutions of the mixed problem for the one-dimensional polytropic filtration equation with inhomogeneous density and source. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time. Note that the technique applied for the proof does not use the Zel'dovich-Kompaneets-Barenblatt solutions, since the construction of such type of function is more complicated in our case. Therefore, we obtain a result by multiplying on a special factor which has convenient properties. In particular, by choosing the parameters of the factor and using the properties of the solution, we obtain the inequality which allows us to show the blow up phenomena. (C) 2015 AIP Publishing LLC.