Asymptotic behavior of the solution of doubly degenerate parabolic equations with inhomogeneous density

In this paper we study the large time behaviour for solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density. Under the suitable assumptions on the data of the problem and on the behaviour of the density at infinity we establish new sharp bound of solutions for a large time. One of the main tool of the proof is new weighted embedding result which is of independent interest. In addition, the proof of uniform estimates of the solution is carried out by modified version of the classical method of De-Giorgi–Ladyzhenskaya–Uraltseva–DiBenedetto. Similar results in the case of power-like density was obtained by one of the author [10]. The approach of this work can be applied for example when studying the qualitative properties of solutions to the Neumann problem for a doubly nonlinear parabolic equation with inhomogeneous density in domains with non-compact boundaries.