Abstract
Consider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.
1 Introduction and the main results
Yin-Wang [1] first studied the equation
where Ω is a bounded domain in RN with appropriately smooth boundary, ρ(x) = dist(x, ∂Ω), p > 1, α > 0. An obvious character of the equation is that, the diffusion coefficient depends on the distance to the boundary. Since the diffusion coefficient vanishes on the boundary, it seems that there is no heat flux across the boundary. However, Yin-Wang [1] showed that the fact might not coincide with what we image. In fact, the exponent α, which characterizes the vanishing ratio of the diffusion coefficient near the boundary, does determine the behavior of the heat transfer near the boundary. One may refer to [1] for the details.
In our paper, we will consider the following equation
The convection term
only a partial boundary condition
matching equation (2) is considered. Here, denoting {ni (x)} as the unit inner normal vector of ∂Ω, when
In [2, 3], we said a bounded domain Ω has the integral non-singularity, if the constants α, p, satisfy
We assumed that there are constants β, c such that
If p > 2, we had obtained the existence of the solution of equation (2) with the initial boundary values (4)–(5), and if Σp = ∂Ω, we also had obtained the stability of the weak solutions. In our paper, we will promote the existence of the solution without the condition (6) but limiting that α ≥ 1. The most innovation of our paper is that the stability of the weak solutions can be obtained only based on the partial boundary condition (4). Comparing with the case of that Σp = ∂Ω in [2, 3] or Σp = ∅ in [1] (when α ≥ p – 1), how to obtain the stability of the weak solutions only based on the partial boundary condition (4) seems very difficult.
Let us give the basic definitions and the main results as following.
A function u(x, t) is said to be a local BV solution of equation(2)with the initial value(5), if
and for any function
The initial value is satisfied in the sense of
Here, QTλ = {(x, t) ∈ QT : ρ(x) = dist(x, ∂Ω) > λ} for small enough λ > 0.
A function u(x, t) is said to be a local BV solution of equation(2)with the initial boundary values(4)–(5), ifu satisfies Definition 1.1, and it satisfies the partial boundary condition(4)in the sense of the trace.
If u is a local BV solution of equation(2), satisfies that
when x is near ∂Ω, then we say u is a regular solution.
Let 1 < p, 1 ≤ α < p – 1, bi(s) ∈ C2(R1). If
then equation(2)with initial boundary values(4)–(5)has a local BV solution u, and ut ∈ L∞(QT).
Let α < p – 1, u and v be two local BV solutions of equation(1)with the same partial homogeneous boundary value
and with the different initial values u(x, 0) = v(x, 0) respectively. If bi(s) is a Lipschitz function, and moreover
then
Here, n > 0 is a nature number, the details of the definition and the properties of the function gn(s) is in Section 3, in particular, |gn(s)s| ≤ c.
Let α ≥ p – 1, and u, v be two local BV solutions of(1)with the initial values u0(x), v0(x) respectively. If u and v are regular, and
then
The most important character of Theorem 1.7 is in that we obtain the stability (15) without any boundary value condition. However, since the solutions considered in the theorem are regular, we can easily obtain the conclusion (15) in a similar way as in [1]. So we omit the details of the proof of the theorem in our paper.
Recently, the author has been interested in the initial-boundary value problem of the following strongly degenerate parabolic equation
The stability of the solutions based on a partial boundary condition (4) has been established in [5–7] et. al. Actually, many mathematicians have been interested in the problem, and have obtained many important results of the the stability of the solutions based on a partial boundary condition, one may see the Refs. [8–11]. Unlike the equation (16), to the best knowledge of the authors, considering the parabolic equation related to the p-Laplacian, our paper is the first one to study the stability of the solutions based on a partial boundary condition (4). Of course, whether the condition (12) in Theorem 1.6 and the assumption that u, v are regular in Theorem 1.7 are necessary or not? This is a very interesting problem to be studied in the future. Some other related references, one can refer to Refs. [12–16]. The paper is arranged as following. In Section 1, we have introduced the problem and given the main results of the paper. In Section 2, we prove the existence of the local BV solution. In Section 3, only based on a partial boundary condition, we prove the Theorem 1.6.
2 The localBV solution
To study equation (2), we consider the following regularized problem
where ρ∊ = ρ * δ∊ + ∊, ∊ > 0, δ∊ is the mollifier as usual. It is well-known that the above problem has a unique classical solution [17, 18]. Hence, for any
If
Proof. By the maximum principle, there is a constant c only dependent on ‖u0‖L∞(Ω) but independent on ε, such that
Multiplying (17) by uε and integrating it over QT, we have
By the fact
then
For small enough λ > 0, let Ωλ = {x ∈ Ω : dist(x, ∂Ω) > λ}. Since p > 1, by (22),
Differentiating (17) with t, and denoting w = u∊t, then
rewriting it as
where
Clearly, w satisfies that
Denoting that
then
By the maximum principle, due to α ≥ 1, we have
By (23)–(24), we know that uε ∈ BV(QTλ), and
Then by Kolmogoroff’s theorem, there exists a subsequence (still denoted as uε) of uε, which is strongly convergent to u ∈ BV(QTλ). In particular, by the arbitrary of λ, uε → u a.e. in QT.
Hence, by (22), (25), there exists a function u and n dimensional vector function
and
In order to prove u satisfies equation (2), we notice that for any function
and uε → u is almost everywhere convergent, so bi(uε) → bi(u) is true. Then
Now, if we can prove that
for any function
Let
By choosing φ = ψu∊ in (26), we can obtain
Noticing that when p ≥ 2,
and when 1 < p < 2,
then in both cases, by (29), we have
Thus
Noticing
and
then
By Hölder inequality, there holds
where
Let ε → 0 in (33). It converges to 0.
Thus, we have
Let v = ψu in (2), we get
Thus
Let v = u – λφ, λ > 0,
If λ → 0, then
Moreover, if λ < 0, similarly we can get
Thus
Noticing that ψ = 1 on suppφ, (28) holds. At the same time, we can prove (5) as in [19], we omit the details here. The lemma is proved. □
If
If α < p – 1, u is a weak solution of equation(1)(also equation(2)) with the initial value(5). Then u has trace on the boundary ∂Ω.
Lemma 2.3 had been proved in [1, 2]. Clearly, Theorem 1.5 is the directly corollary of Lemma 2.1–2.3.
The condition α ≥ 1 is only used to prove that |u∊t| ≤ c, which implies that
3 The stability when α < p – 1
Proof of Theorem 1.6. For a small positive constant λ > 0, let
and let
For any given positive integer n, let gn(s) be an odd function, and
Clearly,
and
where c is independent of n.
By a process of limit, we can choose gn(ϕ(u – v) as the test function, then
Thus
By L’Hospital rule,
Since we assume that ρ|∇u|p < ∞, ρ|∇v|p < ∞, we have
While,
By |bi(u) – bi(v)| ≤ c|u – v|, and by (40), according to the definition of the trace, we have
Moreover,
Since α < p – 1,
In (47), let n → ∞. If {x ∈ Ω : |u – v| = 0} is a set with 0measure, then
If the set {x ∈ Ω : |u – v| = 0} has a positive measure, then,
Therefore, in both cases, we have
Now, after letting λ → 0, let n → ∞ in (41). Then
It implies that
Theorem 1.6 is proved.
Acknowledgement
The paper is supported by NSF of China (no.11371297, 11302184), Natural Science Foundation of Fujian province (no: 2015J01592), supported by Science Foundation of Xiamen University of Technology, China (no:XYK201448).
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