Abstract
In this paper we study the large time behaviour for solutions to the Cauchyproblem for degenerate parabolic equations with inhomogeneous density.Under the suitable assumptions on the data of the problem and on the behaviour of the densityat 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 whichis of independent interest. In addition, the proof of uniform estimatesof the solution is carried out by modified version of the classical methodof De-Giorgi--Ladyzhenskaya--Uraltseva--DiBenedetto. Similar resultsin 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 qualitativeproperties of solutions to the Neumann problem for a doubly nonlinear parabolic equationwith inhomogeneous density in domains with non-compact boundaries.
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