Abstract
The objective of this article is to study the large time asymptotic behavior of the nonnegative weak solution of the following nonlinear parabolic equation with initial condition u(x, 0) = u0(x). By using Moser iteration technique, assuming that the uniqueness of the Barenblatt-type solution E c of the equation u t = div(|Du m |p-2Du m ) is true, then the solution u may satisfy which is uniformly true on the sets . Here B(u m ) = (b1(u m ), b2(u m ), ..., b N (u m )) satisfies some growth order conditions, the exponents m and p satisfy m(p - 1) > 1. Mathematics Subject Classification 2000: 35K55; 35K65; 35B40.
Highlights
The objective of this article is to study the large time asymptotic behavior of the nonnegative weak solution of the nonlinear parabolic equation with the following type ut = div(|Dum|p−2Dum) + div(B(um)), in S = RN × (0, ∞), (1:1)u(x, 0) = u0(x), on RN, (1:2)where m(p - 1) >1, N ≥ 1, u0(x) Î L1(RN), D is the spatial gradient operator, and the convection term div(B(um)) = N i=1 ∂ (bi(um ∂ xi ))Equation (1.1) appears in a number of different physical situations [1].For example, in the study of water infiltration through porous media, Darcy’s linear relation
[5,6]), one obtains with B(s) ≡ 0, u being the volumetric moisture content. Another example where Equation (1.1) appears is the one-dimensional turbulent flow of gas in a porous medium, where u stands for the density, and the pressure is proportional to um-1
According to the different properties of the initial function u0(x), the corresponding nonnegative solutions may have different large time asymptotic behaviors, one can refer to the references [11,12,13,14,15,16,17]
Summary
The objective of this article is to study the large time asymptotic behavior of the nonnegative weak solution of the nonlinear parabolic equation with the following type ut = div(|Dum|p−2Dum) + div(B(um)), in S = RN × (0, ∞),. With B(s) ≡ 0, u being the volumetric moisture content Another example where Equation (1.1) appears is the one-dimensional turbulent flow of gas in a porous medium (cf [7]), where u stands for the density, and the pressure is proportional to um-1 (see [8]). According to the different properties of the initial function u0(x), the corresponding nonnegative solutions may have different large time asymptotic behaviors, one can refer to the references [11,12,13,14,15,16,17]. We are going to study the large time asymptotic behavior for the solution of (1.1) and (1.2) by comparing it to the Barenblatt-type solution, let us give some details
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