Journal of Function Spaces
Volume 2018 (2018), Article ID 6525637, 12 pages
https://doi.org/10.1155/2018/6525637

Research Article

The Well-Posedness of the Solutions Based on the Initial Value Condition

Huashui Zhan¹ and Zhen Zhou²

¹School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China
²School of Science, Jimei University, Xiamen 361021, China

Correspondence should be addressed to Huashui Zhan; huashuizhan@163.com

Received 27 June 2018; Revised 17 August 2018; Accepted 19 August 2018; Published 16 September 2018

Academic Editor: Dumitru Motreanu

Copyright © 2018 Huashui Zhan and Zhen Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The non-Newtonian polytropic filtration equation is considered. Only if , the well-posedness of solutions is studied. If the diffusion coefficient is degenerate on the boundary, then stability of the weak solutions is proved only depending upon the initial value conditions.

1. Introduction

In this paper, we consider the equation

with the initial value condition

but without the usual boundary value condition

where is a bounded domain with smooth boundary, , , , .

This kind of equation is derived from many physical problems [1–5]. In particular, when , , (1) is the evolutionary -Laplacian equation

While, if , , then (1) is porous media equation

Many scholars have made a lot of research on (4) and (5); one can refer to [6–10], etc.

When , (1) becomes usual polytropic infiltration

There are also many papers devoted to this equation. Zhao-Yuan [11] have studied the Cauchy problem of (6) with the initial value ; the existence and the uniqueness of the weak solutions were proved and was shown for any . Fan [12] considered the similar problem when is just a measure.

In [13], Chen-Wnag considered the initial-boundary value problem of the equation

with , . By modifying the usual Morse iteration, imposing some restrictions on the convection function , the local –estimates were made and was obtained. In [14], Tsutsumi considered the initial-boundary value problem of the equation

with the initial value . By showing that , Tsutsumi studied the existence, uniqueness, regularity, and behavior of solutions to (8); we will give more information of [14] at the appendix of this paper. A more general equation was studied by Otto in [15] with the initial value condition. The large time behavior of the solutions of the equations with the type of (1) also has been studied in [16–18], etc. The extinction, positivity, and blow-up of the solutions for a doubly nonlinear degenerate parabolic equation have been studied in [19, 20], etc. Of course, since (1) is one of the most well-known parabolic equations, there are a great deal of papers to study various subjects of this equation; for example, one can refer to [21–27].

If is not a constant, the first thing that catches our attention is when , , , Yin-Wang [28] have made a creative work. They have demonstrated that when , the uniqueness of the weak solution can be proved even if no boundary value condition is required. However, if , based on the initial value (2) and the homogeneous boundary value condition (3), they have proved the uniqueness of the weak solution.

In this paper, we assume , in , on . Different from [28], we only assume that .

Definition 1. Function is a weak solution of problem (1)-(2), if

where .

Theorem 2. Suppose that , , , , , ; then problem (1)-(2) has a solution in the sense of Definition 1, satisfying

where .

When , we can define the trace of (also ) on the boundary. Then the homogeneous boundary value condition (3) can be imposed in the sense of the trace. Nevertheless, the boundary value condition (3) may be redundant.

The main aim of this paper is to prove the stability of the weak solutions of (1) without the boundary value condition (3), no matter whether or not.

Theorem 3. Suppose that , , and are two weak solutions of (1) with different initial value , and ; if

then

where is a small positive constant and .

Theorem 4. Let , , and be two weak solutions of (1) with different initial value , and . If

then

We use some ideas and techniques of [11] to prove Theorem 2. If , , and , the stability theorems of the weak solutions independent of the boundary value condition have been proved by Zhan [16, 17].

2. Prior Estimate

Consider the regularized problem

where , , , and

It is well known that problem (15)-(17) has a nonnegative classical solution , and

Constant C in this paper is independent of , , . Subscripts of are omitted in the following proofs.

Lemma 5. The solution of problem (15)-(17) satisfies that

Proof. Noting that on , multiplying (15) by and integrating it by parts over , we have

Let , we have

Lemma 6. satisfies that

where .

Proof. Multiplying (15) by and integrating it over , we have

that is, (23).
Let ; ; in , ; denote .
By Sobolev inequalities, when ,

By (23) and Lemma 5, we have

Let , , , . Applying also the Hölder inequality, we have

Letting yields (24).
When , by (23) and Sobolev inequalities, (24) still holds.

Lemma 7. satisfies that

where

Proof. Let be the cut-off function of , . Let . Multiplying (15) by and integrating it over , we have

When ,

When ,

By Young’s inequality, when ,

when small enough, since ,

when ,

Let , . By the embedding inequality,

Let , , , and be truncation function of , on . , . We have

By Morse iteration,

that is

By the Young inequality,

from [29], we have

Lemma 8. satisfies that

for any .

Proof. Multiplying (15) by and integrating it over ,

By Lemmas 5–7, is uniformly bounded, so (43) holds.
Multiplying (15) by and integrating it,

Thus, (44) holds.
When integral domain of the above equation is , since , (45) holds.

3. Proof of Theorem 2

3.1.

From Lemmas 5–8, for , , there exist a subsequence of , still denoted by , and a function ; when , we have

where is an open subset such that .

Let . A weak solution of problem (15)-(17), , satisfies

where

Let , .

By Hölder’s inequality and (43)-(48), we have

By (49)

Meanwhile,

By Hölder’s inequality and (48)-(50), we have

and notice that

so

By (49), , a.e. on .

Since , for any , using the Hölder inequality, we have

so , and

By the above discussion, we can deduce that is a weak solution of problem

3.2.

By Lemmas 5–8, is uniformly bounded and uniformly continuous on any compact subset of , so it has a subsequence still denoted by , when ,

From [29] and similar to the previous proof, is a weak solution of problem

3.3.

According to Lemmas 5–8, has a convergent subsequence still denoted by ,

By the Hölder inequality and (43),

that is,

Similar to the previous proof, we have

At last, the initial value condition (2) satisfied in the sense

can be proved in a similar way to that for the evolutionary –Laplacian equation [29]; we omit the details here. Thus is a weak solution of problem (1)-(2).

Lemma 9. If , then .

Proof.

where C is a positive constant independent of .

By this Lemma, we can define the trace of (also ) on the boundary. Thus is a weak solution of the initial-boundary value problem of (1).

4. Proof of Theorem 3

Let ; we have

Let

Denote , , , ; is characteristic function of ().

Let in (71); since , , one has

Since

one has

Let , in (73); we have

that is,

so

Let ; we have

so

Since and are symmetrical,

Let ; we have

5. Proof of Theorem 4

Proof. For a small positive constant , define

as before, where

Then

If and are two weak solutions of (1) with the different initial values and , respectively, we choose as the test function. One has

For the terms on the left hand side of (88),

By the fact that

Using the Young inequality, by (91) (92), one has

which goes to 0 as , since it is assumed that

Now, after letting , let in (88). Then, by (89), (90), and (93), one has

one has

and thus Theorem 4 is proved.

Appendix

The Strong Solution

First of all, let us review the basic definitions and the results of [14]. Consider the initial-boundary value problem of (6), i.e.,

By a weak solution to (A.1)-(2)-(3) we mean a nonnegative function

with

for any , satisfying

and

for any . We call a strong solution to (A.1)-(2)-(3), if is a weak solution and satisfies

If is a strong solution, satisfies (A.1) a.e. in . We call a mild solution if is a weak solution and if there exists a sequence of solution of (A.1) such that for each , with

and

uniformly on every bounded interval as .

Tsutsumi [14] has proved the following theorems.

Theorem A.1. Let be two mild solutions of the initial-boundary value problem (A.1)-(2)-(3) with the same initial data. Then a.e. on .

Theorem A.2. (i) Suppose that   . Then there exists a mild solution to (A.1)-(2)-(3) such that .
(ii) Suppose that and . Then there exists a mild solution to (A.1)-(2)-(3) satisfying for any .
(iii) If , the mild solution in (i), (ii) satisfies for any . If , then .
(iv) If , or , then under the same assumptions as in (i) or in (ii), there exists a strong solution satisfying for any . If , then .

Secondly, if we consider the usual polytropic infiltration equation (6), i.e.,

with the initial-boundary values (2)-(3), (6) is equivalent to (A.1) with that , . By drawing on the weak solution defined in [14] above, we quote the following definition.

Definition A.3. By a weak solution of (6) with the initial-boundary values (2)-(3) we mean a nonnegative function

with

for any , satisfying

and

for any .

Moreover, we can give a version of Theorem A.2 as follows.

Theorem A.4. (i) Suppose that   . Then there exists a mild solution to (6)-(2)-(3) such that .
(ii) Suppose that and . Then there exists a mild solution to (6)-(2)-(3) satisfying for any .
(iii) If , then under the same assumptions as in (i) or in (ii), there exists a strong solution satisfying for any .

At last, we come back to the main equation (1) discussed in this paper. According the basic assumption that , when , we can introduce the following definition.

Definition A.5. By a weak solution of (1) with the initial value (2) we mean a nonnegative function

with

for any , satisfying

and

for any .

We assume that

Consider the asymptotic equation

with the initial-boundary value conditions (2)-(3); similar to the proof of Theorem A.4 above, we know that there is a nonnegative weak solution of (A.19) in the sense of Definition A.3.

Multiplying (A.19) by and integrating it over , we have

Accordingly, by (A.18),

Multiplying (A.19) by and integrating it over ,

Noticing that

then

Combining (A.22)-(A.24) and

we deduce that

which implies that

By (A.20) and (A.27), using the usual weak convergence method, we are able to prove the following theorem.

Theorem A.6. Suppose that satisfies (A.18); then there exists a weak solution of (1) with the initial value (2) in the sense of Definition A.5. Moreover, if , then there is a weak solution of (1) with the initial-boundary value (2)-(3), where the boundary value (3) is true in the sense of the trace.

Now, one can see that Theorem 2 in the introduction is better than Theorem A.6. This is the more reason we use the idea and the method of [11], rather than the method of [14], to study the existence of the solution to (1).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The paper is partially supported by NSF of Fujian Province and partially supported by NSF of Xiamen University of Technology, China.

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