The boundary degeneracy of a singular diffusion equation
© Zhan and Xie; licensee Springer. 2014
Received: 28 October 2013
Accepted: 14 July 2014
Published: 18 August 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.
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.
then the function u is said to be a weak solution of (1.1).
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.
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 .
where is the closure of in space of .
holds, where , and .
By the above two lemmas, we are able to get the following theorem.
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.
Denote , and for simplicity, denote as u.
Assume that is the cut off of functions smoothly in , , , , , , .
where ε is an appropriately small positive constant.
Then we obtain the theorem.
Also we can obtain Theorem 2.3 according to (2.5).
The proof of Theorem 2.3 is complete. □
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.
here and in what follows c is a constant independent of ε.
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.
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.
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.
where we have used the facts that , .
Thus (3.5) holds in both cases. □
Here the constant makes the inclusion true as before.
Now, we estimate u below, near the boundary ∂ Ω.
The main idea is to construct a subsolution of the equation.
where we have used the fact that .
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.
However, we do not use this estimate in the following proof.
The theorem is proved. □
The paper is supported by NSF of China (no. 11371297), supported by SF of Xiamen University of Technology, China.
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