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(|Dum|p-2Dum) is true, then the solution u may satisfy
which is uniformly true on the sets . Here B(um) = (b1(um), b2(um), ..., b N (um)) satisfies some growth order conditions, the exponents m and p satisfy m(p - 1) > 1.
Mathematics Subject Classification 2000: 35K55; 35K65; 35B40.
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1. Introduction
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
where m(p - 1) > 1, N ≥ 1, u0(x) ∈ L1(RN), D is the spatial gradient operator, and the convection term .
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
satisfactorily describes flow conditions provided the velocities are small. Here V represents the seepage velocity of water, θ is the volumetric moisture content, K(θ) is the hydraulic conductivity and ϕ is the total potential, which can be expressed as the sum of a hydrostatic potential ψ(θ) and a gravitational potential z
However, (1.3) fails to describe the flow for large velocities. To get a more accurate description of the flow in this case, several nonlinear versions of (1.3) have been proposed. One of these versions is
where α ranges from 1 for laminar flow to 2 for completely turbulent flow (cf. [2–4] and references therein). If it is assumed that infiltration takes place in a horizontal column of the medium, according to the continuity equation
then (1.4) and (1.5) give
with and D(θ) = K(θ)ψ'(θ). Choosing D(θ) = D0θm-1(cf. [5, 6]), one obtains (1.1) 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 also [8]). Typical values of p are again 1 for laminar (non-turbulent) flow and for completely turbulent flow.
The existence of nonnegative solution of (1.1)-(1.2) without the convection term div(B(um)), defined in some weak sense, had been well established (see [9] etc.). Here we quote the following definition.
Definition 1.1. A nonnegative function u(x, t) is called a weak solution of (1.1)-(1.2) if u satisfies
(i)
(ii)
(iii)
If there exist the positive constants k1, α such that
Chen-Wang [10] had proved the existence and the uniqueness of the weak solutions of (1.1) and (1.2) in the sense of Definition 1.1.
As we have said before, we are mainly interested in the behavior of solution of (1.1) and (1.2) as t → ∞. 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 [11–17]. In our article, 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.
It is not difficult to verify that
is the Barenblatt-type solution of the Cauchy problem
where , b is a constant such that , and δ denotes the Dirac mass centered at the origin.
By using some ideas of [9, 14], we have the following
Theorem 1.2. Suppose m(p - 1) > 1, B satisfies (1.10) with α < p - 1 and . If E c is a unique solution of (1.11) and (1.12), then the solution u of (1.1) and (1.2) satisfies
uniformly on the sets , where as before.
Remark 1.3. For m = 1, the uniqueness of solutions of (1.11) and (1.12) is known (see [18]).
By assuming that the uniqueness of the Barenblatt-type solution of (1.11) is true, Yang and Zhao [14] had established the similar large time behavior of solution of the Cauchy problem of the following equation
While Zhan [17] had considered the Cauchy problem of the following equation
and also had got the similar result as Theorem 1.2. Comparing (1.1) with (1.13) or (1.14), the most difficulty comes from that the convection term div(B(um)). The absorption term -uq in (1.13), or in (1.14), is always less than 0. This fact made us be able to draw it away in many estimates in [14] or [17]. But the convection term div(B(um)) plays important role in this article, and it can not be drawn away randomly in the estimates we needed, we have to deal with it by some special techniques.
At the end of this introduction section, we would like to point that the condition m(p - 1) > 1 in Theorem 1.2, which means that the Equation (1.1) or (1.11) is a doubly degenerate parabolic equation, plays an important role in the proof of the theorem. In other words, if it is not true, (1.1) is in singular case, then the large time behavior of the solution in this case is still an open problem.
2. Some important lemmas
Let u be a nonnegative solution of (1.1) and (1.2). We define the family of functions
It is easy to see that they are the solutions of the problems
where as before and u0k(x) = kN u0(kx).
Lemma 2.1 For any s ∈ (mα, m(p - 1)), the nonnegative solution u k satisfies
Proof. From Definition 1.1, we are able to deduce that (see [19]): for , φ = 0 when |x| is large enough, for any t ∈ [0, T], 0 < h < t,
Let
By an approximate procedure, we can choose in (2.5), then
Noticing
Since s > αm, ,
is always true, we have
and
Noticing that the condition m(p - 1) > 1 and
then by (2.7)-(2.11), we obtain
Since u k ∈ L∞ (RN × (h, T)) ∩ L1(S T ),
Let h → 0 in (2.12). Then
From this inequality, it is clear of that
So (2.3) is true.
Let
By Sobolev's imbedding inequality (see [19]), for , ξ ≥ 0, we have
where
It follows that
where we denote S T = RN × (0, T). Since
we have
Hence, by (2.16), (2.(17) and (2.15), we get
Let , ψ R . be the function satisfying (2.6) and . Then
by Moser iteration technique, the above inequality implies (2.4) is true.
Let Q ρ = B ρ (x0) × (t0 - ρp, t0) with t0 > (2ρ)p and uk 1= max{u k , 1}. Also by Moser iteration technique, we have
Lemma 2.2 The nonnegative solution u k satisfies
where c(ρ, s1) depends on ρ and s1, and s1 can be any number satisfying .
Proof. For , φ = 0 when |x| is large enough, we have
Let ξ be the cut function on Q ρ , i.e.
We choose the testing function in (2.19) as , where is a constant. Then
Using Young inequality, by (1.10),
from (2.20), we have
By the fact of that
from (2.21), we have
Let
and
By the embedding theorem, from (2.22), we have
where
In particular, we choose
then from (2.23), we have
Now, for , we denote that
and choose the cut functions ξ l (x, t) of Q ρl , such that on Qρ(l+1), ξ l = 1.
Denote
and let
Then, by (2.23) and (2.24) and the assumption of that a < p - 1, which implies
we have
Using Moser iteration technique, we have
Then, we have
By Schwarz inequality,
By the Lemma 3.1 in [19], for any , we have
and from this inequality, we get the conclusion of the lemma.
Lemma 2.3 The nonnegative solution u k satisfies
Proof. By Lemmas 2.1 and 2.2, {u k } are uniformly bounded on every compact set K ⊂ S T . Let ψ R be a function satisfying (2.6) and with 0 ≤ ξ ≤ 1, ξ = 1 if t ∈ (τ, T). We choose in (2.5) to obtain
Noticing
from these inequalities, by (2.18) and (2.27), one knows that (2.25) is true. (2.26) is to be proved in what follows.
Let
Then
By (2.1) and (2.29), for any , we have
Let g n (s) = 1 when ; g n (s) = ns when ; g n (s) = 0 when s < 0, and let φ in (2.31) be substituted by . Then
Let . Where , 0 ≤ θ ≤ 1, θ(x) = 1 when x ∈ B1, and , 0 ≤ η j ≤ 1, which satisfies that η j → η when j → ∞, and η is the characteristic function of (s1, s2), s1 < s2.
Since u k , v ∈ L∞ (RN × (τ, T)), , Dvm ∈ Lp (RN × (τ, T)), we have
as k → ∞.
If we notice that, for any i ∈{1,2, ..., N},
then it is easy to show that
At the same time,
Then, if we let k → ∞, n → ∞ and let r → 1 in (2.32), since μ < α , we have
in other words,
Let j → ∞. Then
Similarly, we have
Let s1 → 0. Then
It follows that
which implies that
Denote w = tγu k (x, t), . By (2.34), w t ≥ 0. By (2.1),
From (2.15), (2.18) and (2.35), we obtain (2.26).
3. Proof of Theorem 1.2
Proof of Theorem 1.2. By Lemmas 2.1-2.3, there exists a subsequence of {u k } and a function v such that on every compact set K ⊂ S
Similar to what was done in the proof of Theorem 2 in [9], we can prove v satisfies (1.11) in the sense of distribution.
We now prove v(x, 0) = cδ(x). Let . Then we have
To estimate , without loss of the generality, one can assume that u k > 0. By Hölder inequality and Lemma 2.1,
where , uk 1= max(u k , 1).
Hence from (3.1), we get
Letting k → ∞, t → 0 in turn, we obtain
Thus
v(x, t) is a solution of (1.11) and (1.12). By the assumption on uniqueness of solution, we have v(x, t) = E c (x, t) and the entire sequence {u k } converges to E c as k → ∞. Set t = 1.
Then
uniformly on every compact subset of RN. Thus by writing kx = k', kNμ = t' , and dropping the prime again, we see that
uniformly on the sets , a > 0. Thus Theorem 1.2 is true.
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Acknowledgements
The article is supported by NSF (no. 2009J01009) of Fujian Province, Pan Jinlong's SF of Jimei University in China.
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Zhan, H., Xu, B. The asymptotic behavior of the solution of a doubly degenerate parabolic equation with the convection term. J Inequal Appl 2012, 120 (2012). https://doi.org/10.1186/1029-242X-2012-120
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DOI: https://doi.org/10.1186/1029-242X-2012-120