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Journal of Function Spaces
Volume 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

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; moc.361@nahziuhsauh

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 equationwith the initial value conditionbut without the usual boundary value conditionwhere is a bounded domain with smooth boundary, , , , .

This kind of equation is derived from many physical problems [15]. In particular, when , , (1) is the evolutionary -Laplacian equationWhile, if , , then (1) is porous media equationMany scholars have made a lot of research on (4) and (5); one can refer to [610], etc.

When , (1) becomes usual polytropic infiltrationThere 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 equationwith , . 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 equationwith 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 [1618], 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 [2127].

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, satisfyingwhere .

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 ; ifthenwhere is a small positive constant and .

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

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 problemwhere , , , andIt is well known that problem (15)-(17) has a nonnegative classical solution , andConstant 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 haveLet , we have

Lemma 6. satisfies thatwhere .

Proof. Multiplying (15) by and integrating it over , we havethat is, (23).
Let ; ; in , ; denote .
By Sobolev inequalities, when ,By (23) and Lemma 5, we haveLet , , , . Applying also the Hölder inequality, we haveLetting yields (24).
When , by (23) and Sobolev inequalities, (24) still holds.

Lemma 7. satisfies thatwhere

Proof. Let be the cut-off function of , . Let . Multiplying (15) by and integrating it over , we haveWhen ,When ,By Young’s inequality, when ,when small enough, since ,when ,Let , . By the embedding inequality,Let , , , and be truncation function of , on . , . We haveBy Morse iteration,that is By the Young inequality, from [29], we have

Lemma 8. satisfies thatfor any .

Proof. Multiplying (15) by and integrating it over ,By Lemmas 57, 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 58, for , , there exist a subsequence of , still denoted by , and a function ; when , we havewhere is an open subset such that .

Let . A weak solution of problem (15)-(17), , satisfieswhereLet , .By Hölder’s inequality and (43)-(48), we haveBy (49)Meanwhile,By Hölder’s inequality and (48)-(50), we haveand notice thatsoBy (49), , a.e. on .

Since , for any , using the Hölder inequality, we haveso , andBy the above discussion, we can deduce that is a weak solution of problem

3.2.

By Lemmas 58, 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 58, has a convergent subsequence still denoted by ,By the Hölder inequality and (43),that is, Similar to the previous proof, we haveAt last, the initial value condition (2) satisfied in the sensecan 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 haveLet

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

Let in (71); since , , one hasSince one has

Let , in (73); we have

that is, soLet ; we havesoSince and are symmetrical,Let ; we have

5. Proof of Theorem 4

Proof. For a small positive constant , define