Mathematics & Physics 2016, 9(2), 225–234 УДК 517.957 On the Properties of Solutions of Multidimensional Nonlinear Filtration Problem with Variable Density and Nonlocal Boundary Condition in the Case of Fast Diffusion Zafar R.Rakhmonov∗ National University of Uzbekistan 100174, University street, 4. <...> Tashkent Uzbekistan Received 30.08.2015, received in revised form 04.12.2015, accepted 12.01.2016 The conditions of global existence of solutions of a nonlinear filtration problem in an inhomogeneous medium are investigated in this paper. <...> Various techniques such as the method of standard equations, selfsimilar analysis and the comparison principle are used to obtain results. <...> The influence of inhomogeneous medium on the evolution process is analyzed. <...> The critical global existence exponent and the critical Fujita exponent are obtained. <...> Asymptotic behavior of solutions in the case of the global solvability is established. <...> For example, equation (1) arises in mathematical modelling of reaction-diffusion process in nonlinear media, fluid flows through porous media, dynamics of biological populations, polytropic filtration, synergy structures and various other phenomena [2, 4]. <...> Equation (1) is a parabolic equation with inhomogeneous density [2]. <...> They also found that solutions of problem (1)–(3) have the following properties: data. ∗zraxmonov@inbox.ru • if 0 < q 2(p−1)/p then global solution of problem (1)–(3) exists; • if q > 2(p − 1) then problem (1)–(3) admits nontrivial global solutions with small initial ©Siberian Federal University. <...> Similar results were established for various nonlinear parabolic equations [3, 4, 8, 14, 15]. <...> In the case of slow diffusion it has been proved that solution Here q0 = 2(p−1)/p is the critical global existence exponent and qc = 2(p−1) is the critical of problem (1)–(3) is global in time when 0 < q (m+1)(p−1)/p. <...> Moreover, they also proved that qc = (m+ 1)(p − 1) is the critical Fujita exponent. <...> If (m+ 1)(p − 1)/p < q < qc then all solutions become unbounded in finite time but if q > qc global solutions <...>