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The boundary degeneracy of a singular diffusion equation
Journal of Inequalities and Applications volume 2014, Article number: 284 (2014)
We consider the following singular diffusion equation with boundary degeneracy: , , where is a bounded domain with appropriately smooth boundary, , , and . Though its diffusion coefficient vanishes on the boundary, it is still possible that the heat flux transfers across the boundary (Yin and Wang in Chin. Ann. Math., Ser. B 25:175-182, 2004), and it is not possible to define the homogeneous boundary value condition as usual. In the paper, under the assumption on the uniqueness of the weak solution, if the point x lies in the interior of the domain Ω, the paper obtains the result that the weak solution of the quoted equation has the same regular properties as those of the weak solution to the usual evolutionary p-Laplacian equation. However, if the point x lies on the boundary ∂ Ω, the situation may be different. The most significant feature of the paper is that the definition of the homogeneous boundary value condition of the above equation is given. Then, if , the bounded estimates of the weak solution are got by constructing the special barrier functions, and at last, how the diffusion coefficient affects the gradient of the solution near the boundary is discussed.
MSC:35K55, 35K65, 35B40.
Consider the following singular diffusion equation with boundary degeneracy:
where is a bounded domain with appropriately smooth boundary, , , and is the distance function from the boundary. If , then (1.1) becomes the following evolutionary p-Laplacian equation:
Equation (1.2) reflects the more practical process of heat conduction than the classical heat conduction equation does. For example, when , the solution of the equation may possess the property of propagation of finite speed, while always has the property of propagation of infinite speed which seems clearly contrary to the practice. There is a tremendous amount of related work for (1.2), one is referred to the books [1–3]etc. and the references therein.
For (1.1), 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,  has shown that the fact might not coincide with what we imagine to be the case. 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. Let us give the definition of weak solution for (1.1) as follows.
Definition 1.1 If the function satisfies , , , and for any test function , the following integral equality holds:
then the function u is said to be a weak solution of (1.1).
According to , if , we can impose the Dirichlet boundary value condition as usual
Otherwise, if , then the heat conduction of (1.1) is entirely free from the limitations of the boundary condition. In other words, the problem of heat conduction is entirely controlled by the initial value condition
This fact makes us consider whether the properties of the solutions, such as the regularity, the large time behavior etc. of (1.1) are the same as the corresponding properties of the solutions of (1.2) or not.
In this paper, under the assumption of the uniqueness of the solution to the singular diffusion (1.1), Section 2 discusses the regular properties of the solution of (1.1) in the interior points of by using the method as the Chapter 2 of . Section 3 first gives the definition of the homogeneous boundary value condition of (1.1), then by using some ideas of , if , the boundedness estimates of the weak solution of (1.1) on the boundary are obtained. In the last section of the paper, we emphasize the analysis of how the diffusion coefficient affects the gradient of the solution near the boundary in some cases.
Remark 1.2 If the initial value satisfies
then the results of  have shown that the uniqueness of the solution to (1.1) is true, only if . The existence and the uniqueness of the solution to a more general equation than (1.1) had been studied in .
2 Estimate in the interior of the domain
We first introduce the following lemmas from .
Lemma 2.1 There exists a constant γ only depending on p, q, N such that for any and ,
where is the closure of in space of .
Lemma 2.2 Let () be a sequence of bounded open sets in . Assume for any , and that there exist some constants , , , such that the following inequality holds:
holds, where , and .
By the above two lemmas, we are able to get the following theorem.
Theorem 2.3 Let Ω be a uniformly domain. If u is the unique solution of (1.1) in , and , then
The proof of the theorem is just similar to that Proposition 4.1 in Chapter 2 of , in which the same conclusion on (1.2) is obtained.
Proof Since u is the unique solution of (1.1) in , we can assume that u is the limit of , which is the classical solution of the following regularized problem:
where is the smoothly mollified functions of . Let be an any compact set. Similar to the proof of Lemma 2.3 in the Chapter 2 of , which discusses (1.2), we are able to get
Denote , and for simplicity, denote as u.
Now, we differentiate (2.2) with respect to , and we obtain
Assume that is the cut off of functions smoothly in , , , , , , .
Let , . We choose , multiply the two sides of (2.6) with , and integrate on . Then we can obtain the following equality:
Using the fact of that, according to , the distance function always satisfies in the sense of distribution, and clearly
By the Young inequality,
where ε is an appropriately small positive constant.
(1) When , denote
From (2.9), we can obtain
by Lemma 2.1 and (2.10),
which implies that
where . By choosing , the above formula can be changed into
then, by the Hölder inequality, we see that
which implies that
Then we obtain the theorem.
If (2.12) is not true, we have
By Lemma 2.2,
where is a positive integer which makes hold. Then we can obtain Theorem 2.3 according to (2.5). Therefore, when ,
(2) When , we can get
Then according to (2.9),
where . Then using Lemma 2.1, similar to the discussion of the case (1), we can obtain
then by the Hölder inequality, we have
which implies that
If (2.16) is not true, then from (2.15), we have
Then using Lemma 2.2, we obtain
Also we can obtain Theorem 2.3 according to (2.5).
Therefore when , also
The proof of Theorem 2.3 is complete. □
Theorem 2.4 Supposed that u is a weak solution of (1.1) in , then for any compact set , ,
where c is a constant only dependent on N, p, , .
The proof of the theorem is just similar to that of Theorem 4.3 in Chapter 2 of , in which the same conclusion on (1.2) is obtained.
Proof We only need to prove that u satisfies (2.17) in , for any such that , . Let be the usual mollified function of u,
where . Then for any , according to the definition of , we easily get
According to Theorem 2.3, we have
here and in what follows c is a constant independent of ε.
Now, let , , , , . Then, similar to Chapter 2 of , we have
For fixed , , . Let us choose the test function in the definition of generalized solution (1.3) as . Then we have
By (2.19), we obtain
Let , , , when , . For any , we define . By the approximation process, we know that (2.20) is also true for any . Choosing in (2.20), then
We notice that, when , . But holds when ; then and
Therefore using Theorem 2.3 and (2.21), letting , we obtain
By the mean value theorem, there exists , such that
combining this formula with (2.18), letting , we obtain the desired result. □
By Theorems 2.3 and 2.4, similar to Chapter 2 of , it is not difficult to prove the following theorem, and we omit the details here.
Theorem 2.5 Let , u is the generalized solution of (1.1) in , then () is locally Hölder continuous in .
3 Estimates on the boundary
Now, we assume that the boundary ∂ Ω is of class . That is, there exists a number such that for all the portion of ∂ Ω within the ball can be represented, in a local system of coordinates, as the graph of a function such that , and for , .
Let u be the unique nonnegative bounded solution of (1.1) in the sense of Definition 1.1. Then satisfies , , . From this definition, we know that one cannot define the trace of u on the boundary except for .
But the results of  show that if , one can define the trace of u on the boundary, and the homogeneous boundary value condition can be defined as usual. However, if , then the heat conduction of (1.1) is entirely free from the limitations of the boundary condition, in other words, the problem of heat conduction is entirely controlled by the initial value condition, then in this case one cannot give the homogeneous boundary value condition as usual. Fortunately, no matter how the diffusion coefficient α satisfies or , the results of  had shown that the uniqueness of the solution is true (cf. Remark 1.2) only if the initial value is suitably smooth. Then we can give the following definition.
Definition 3.1 If is suitably smooth, u is the limit of the solutions of the following problem:
where is the smoothly mollified functions of . Then we say u is the solution of (1.1) with the homogeneous boundary value condition.
We will get Estimates above.
Theorem 3.2 Let u be the unique nonnegative bounded solution of (1.1) with the homogeneous boundary value condition in the sense of Definition 3.1. Supposed that , for ,
where the constant C depending upon M, N, p, s, and the constant k is a constant independent of s, M.
The key idea of the proof is to work with cylinders whose dimensions are suitably rescaled to reflect the degeneracy exhibited by the equation. The main idea is to construct a supersolution of the equation.
Proof Fix . We may assume that and in the vicinity of , after flattening of ∂ Ω near , without loss of generality, let us assume that ∂ Ω coincides with the portion of hyperplane , the inclusion . Let , and the set
We assume k is so large that . Consider the following problem:
where is the solution of the problem (3.1)-(3.3), , is small enough, and . By the comparison theorem [, p.119], we have
Let us construct a barrier for u in . Consider the function
Our barrier is given by
where the constants γ, C are to be chosen later so large that on the parabolic boundary of . This holds true on the portion of such a boundary lying on the hyperplane ,
provided that . On the portion , we have
if . On the bottom of we have
By direct calculation,
where we have used the facts that , .
Clearly, if we choose k large enough, then we have
It follows by the comparison principle that the solution of the problem (3.6)-(3.8) v, in . In particular, , due to the interior regularity of the solution of (1.1), which we have discussed in Section 2, we have
Therefore there exists a constant k depending only upon N, such that
for all such that . On the other hand, if , we have
Thus (3.5) holds in both cases. □
Estimates below: Let u be a nonnegative bounded solution of (1.1) with homogeneous boundary value condition in the sense of Definition 3.1,
for some . For let
For , let
Here the constant makes the inclusion true as before.
Now, we estimate u below, near the boundary ∂ Ω.
Theorem 3.3 If the hypothesis of Theorem 3.2 is true, then for , , , the inequality
The main idea is to construct a subsolution of the equation.
Proof Fix and let . After flattening of ∂ Ω near , we may assume that as before. Introduce the point
and the domain
where is the nonnegative solution of the problem (3.1)-(3.3), is small enough, and . Also by the comparison theorem [, p.119], we have
Consider the function
and construct the barrier
where is large enough constant chosen later. Let us show that on the parabolic boundary of . On the portion we have . On the portion lying on the hyperplane one checks that . On the bottom of , we have
By direct calculation
where . We have
where we have used the fact that .
Clearly, if we choose γ large enough, then we have
It follows from the comparison principle that in . In particular, ,
also, due to the interior regularity of the solution (1.1), which we have discussed in Section 2, such that , let . We have
Theorem 3.3 is proved. □
Remark 3.4 The method of estimates near the boundary we used here is classical, which strongly depends on the construction of special barrier functions. The shortcoming of such a technique is evident even in the framework of an evolutionary p-Laplacian equation (1.2) itself (see etc.). For the diffusion (1.1) with , whether the estimate (3.5) or (3.10) is true or not is a problem to be probed in the future. To solve this open question, it strongly depends on how to extend the Harnack estimates to the case of parabolic equations with the full quasilinear structure, as it happens when . Results of this kind would probably require a new method independent of local representations and local subsolutions. Whenever developed, such a technique may parallel the discovery of the Moser estimates , based on real and harmonic analysis tools, versus the estimates by Hadamards  and Pini , based on local representations.
4 Estimates of the gradient near the boundary
In the last section, we are concerned with the estimates of the gradient of the solution near the boundary. We will prove the following theorem.
Theorem 4.1 Supposed that , . Supposed that the points is close enough to the boundary ∂ Ω, then for the unique weak solution u of (1.1) with the homogeneous boundary value in the sense of Definition 3.1, there exists a constant γ, such that for all , is small enough, we have
Proof of Theorem 4.1 For , consider the cylindrical domain
Since we are concerned with those points near the boundary, without loss of generality, we may assume is small enough such that
and that . Introduce the change of variables
and the new function
This maps into the box . Moreover, ψ satisfies
in which , but it is not the distance function of to the boundary of . If , in such a box, according to Theorems 2.3-3.3, we have
However, we do not use this estimate in the following proof.
Similar to the discussion of (2.10) in Theorem 2.3, for simplicity, we still denote as in the following, and we get
where , , . By Lemma 2.1 and (4.9),
which implies that
where . By choosing , the above formula can be changed into
then by the Hölder inequality, we see that
which implies that
If (4.12) is not true, we have
By Lemma 2.2,
where is a positive integer which makes hold. Then, according to (2.5), which is still true, and because the constant C is independent of , we have
Especially, we choose , , from (4.13)-(4.14), we have
Now in terms of u, it means that
The theorem is proved. □
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The paper is supported by NSF of China (no. 11371297), supported by SF of Xiamen University of Technology, China.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Zhan, H., Xie, Q. The boundary degeneracy of a singular diffusion equation. J Inequal Appl 2014, 284 (2014). https://doi.org/10.1186/1029-242X-2014-284
- boundary degeneracy
- diffusion equation
- regular property
- barrier function