Mathematics & Physics 2015, 8(2), 192–200 УДК 517.955.8 The Properties of the Solutions for Cauchy Problem of Nonlinear Parabolic Equations in Non-Divergent Form with Density Jakhongir R.Raimbekov∗ National University of Uzbekistan Yunus Abad-17, 3/66, 10037 Tashkent, Uzbekistan Received 02.04.2014, received in revised form 15.10.2014, accepted 03.04.2014 We investigate the solutions for the following nonlinear degenerate parabolic equation in non-divergent form with density |x|n ∂u ∂t = umdiv |∇u|p−2∇u We discuss the properties, which are different from those for the equations in divergence form, thus generalizing various known results. <...> Then getting a self-similar solution we show the asymptotic behavior of solutions at t → ∞. <...> Finally, we present the results of some numerical experiments. <...> Keywords: nonlinear degenerate parabolic equation, non-divergent form, self-similar solution, asymptotic behavior of solutions. <...> The equation (1) describes many physical problems such as dispersal mechanisms on specials not expect to have a classical solution in general, we consider only weak solutions which are nonnegative and in the following weak sense. <...> The equation (1) may be degenerate at the points where u = 0 and ∇u = 0. <...> The cases m = 1 or p = 2 (i.e. p-Laplacian equation and porous medium equation) were thoroughly studied by many authors (see e.g. [9,10,18]). diffusion equation with double nonlinearity in divergent form, which has been studied in [1]. <...> Giving the requirements for the initial values and defining the solutions in a specific way they showed a property of uniqueness for the viscosity solutions which is missing for classic and even other weak solutions. <...> Dal Passo and Luckhaus claim that for the case p = 2 uniqueness fails since for every T > 0 a weak solution with extinction time T. They just got a unique maximal solution and showed that its support remains constant [5]. <...> However, in [12] a counter example was shown by Ughi, who has proved uniqueness of the. <...> The support of solutions of the equation (1) will never expand atm > 1, while it is known that the equations in divergence form have the property of the finite (or infinite) speed <...>