The decay rate of the solution to the Cauchy problem for doubly nonlinear parabolic equation with absortion

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.