Abstract:
This work deals with the Cauchy problem for a wide class of quasilinear
second-order degenerate parabolic equations with inhomogeneous
density and absorption terms. It is well known that for the problem
under consideration but without absorption term and when the density
tends to zero at infinity not very fast the mass conservation law
holds true. However that fact is not always valid with an absorption
term. In this paper, the precise conditions on both
the structure of nonlinearity and inhomogeneous density which
guarantee the decay to zero of the total mass of solution as time
goes to infinity is established. In other words the criteria of stabilization to
zero of the total mass for a large time is established in terms of
critical exponents. As a consequence of obtained results and local Nash-Mozer
estimates the sharp sup bound of a solution is done as well.