The Well-Posedness of the Solutions Based on the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06569958pt" id="M1" height="15.3664pt" version="1.1" viewBox="-0.0657574 -15.3007 16.1859 15.3664" width="16.1859pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g><g transform="matrix(.012,0,0,-0.012,9.484,-7.578)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg> Initial Value Condition

Zhan, Huashui; Zhou, Zhen

doi:https://doi.org/10.1155/2018/6525637

Journal of Function Spaces

On this page

Abstract Introduction Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 6525637 | https://doi.org/10.1155/2018/6525637

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

Huashui Zhan¹and Zhen Zhou²

Academic Editor: Dumitru Motreanu

Received27 Jun 2018

Revised17 Aug 2018

Accepted19 Aug 2018

Published16 Sept 2018

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 [1–5]. 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 [6–10], 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 [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, 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 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 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 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 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 , defineas 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 hasFor the terms on the left hand side of (88),By the fact thatUsing the Young inequality, by (91) (92), one haswhich goes to 0 as , since it is assumed that Now, after letting , let in (88). Then, by (89), (90), and (93), one has one hasand 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 withfor any , satisfyingandfor 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 , withand 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 withfor any , satisfyingandfor 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 functionwithfor any , satisfyingandfor any .

We assume that

Consider the asymptotic equationwith 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 haveAccordingly, by (A.18),

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

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

References

A. S. Kalashnikov, “Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,” Uspekhi Mat Nauk, vol. 254, no. 2, pp. 135–176, 1987.
View at: Google Scholar | MathSciNet
R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon, Oxford, UK, 1975.
View at: Publisher Site
J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, NY, USA, 1972.
View at: Publisher Site
E. C. Childs, An Introduction to the Physical Basis of Soil Water Phenomena, Wiley, London, UK, 1969.
View at: Publisher Site
L. A. Richards, “Capillary conduction of liquids through porous mediums,” Journal of Applied Physics, vol. 1, no. 5, pp. 318–333, 1931.
View at: Publisher Site | Google Scholar
E. Di Benedetto, Degenerate Parabolic Equations, Springer, New York, NY, USA, 1993.
View at: Publisher Site | MathSciNet
E. Di Benedetto, “On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 13, no. 4, pp. 487–535, 1986.
View at: Google Scholar | MathSciNet
G. Lu, “Nonlinear degenerate parabolic equations in infiltration through a porous medium,” Communications in Nonlinear Science & Numerical Simulation, vol. 3, no. 2, pp. 97–100, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
B. E. Dahlberg and C. E. Kenig, “Nonnegative solutions of the porous medium equation,” Communications in Partial Differential Equations, vol. 9, no. 5, pp. 661–678, 1984.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Ohara, “L1 estimates of solutions of some nonlinear degenerate parabolic equations,” Nonlinear Anal. TMA, vol. 18, no. 5, pp. 413–426, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
J. N. Zhao and H. J. Yuan, “The Cauchy problem for a class of nonlinear doubly degenerate parabolic equations,” Chinese Annals of Mathematics. Series A, vol. 16, no. 2, pp. 181–196, 1995.
View at: Google Scholar | MathSciNet
H. J. Fan, “Cauchy problem of some doubly degenerate parabolic equations with initial datum a measure,” Acta Mathematica Sinica, vol. 20, no. 4, pp. 663–682, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
C. S. Chen and R. Y. Wang, “Global existence and L1 estimates of solution for doubly degenerate parabolic equation,” Acta Mathematica Sinica, vol. 44, no. 6, pp. 1089–1098, 2001.
View at: Google Scholar | MathSciNet
M. Tsutsumi, “On solutions of some doubly nonlinear degenerate parabolic equations with absorption,” Journal of Mathematical Analysis and Applications, vol. 132, no. 1, pp. 187–212, 1988.
View at: Publisher Site | Google Scholar | MathSciNet
F. Otto, “L1 -contraction and uniqueness for quasilinear elliptic-parabolic equations,” Journal of Differential Equations, vol. 131, no. 1, pp. 20–38, 1996.
View at: Publisher Site | Google Scholar | MathSciNet
H. Zhan, “Large time behavior of solutions to a class of doubly nonlinear parabolic equations,” Applications of Mathematics, vol. 53, no. 6, pp. 521–533, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
H. Zhan, “The asymptotic behavior of solutions for a class of doubly degenerate nonlinear parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 370, no. 1, pp. 1–10, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
D. Andreucci, G. R. Cirmi, S. Leonardi, and A. F. Tedeev, “Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary,” Journal of Differential Equations, vol. 174, no. 2, pp. 253–288, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
H. J. Yuan, S. Z. Lian, C. L. Cao, W. J. Gao, and X. J. Xu, “Extinction and positivity for a doubly nonlinear degenerate parabolic equation,” Acta Mathematica Sinica, vol. 23, no. 10, pp. 1751–1756, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
A. F. Tedeev, “The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations,” Applicable Analysis: An International Journal, vol. 86, no. 6, pp. 755–782, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Zhou, Z. Guo, and B. Wu, “A doubly degenerate diffusion equation in multiplicative noise removal models,” Journal of Mathematical Analysis and Applications, vol. 458, no. 1, pp. 58–70, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
J. Sun, J. Yin, and Y. Wang, “Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation,” Nonlinear Analysis, vol. 74, no. 6, pp. 2415–2424, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
R. Gianni, A. Tedeev, and V. Vespri, “Asymptotic expansion of solutions to the Cauchy problem for doubly degenerate parabolic equations with measurable coefficients,” Nonlinear Analysis, vol. 138, pp. 111–126, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
H. Shang and J. Cheng, “Cauchy problem for doubly degenerate parabolic equation with gradient source,” Nonlinear Analysis, vol. 113, pp. 323–338, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
J. Droniou, R. Eymard, and K. S. Talbot, “Convergence in of weak solutions to perturbed doubly degenerate parabolic equations,” Journal of Differential Equations, vol. 260, no. 11, pp. 7821–7860, 2016.
View at: Publisher Site | Google Scholar
W. Zou and J. Li, “Existence and uniqueness of solutions for a class of doubly degenerate parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 446, no. 2, pp. 1833–1862, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Q. Li, “Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations,” Nonlinear Analysis, vol. 170, pp. 88–122, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
J. Yin and C. Wang, “Properties of the boundary flux of a singular diffusion process,” Chinese Annals of Mathematics, Series B, vol. 25, no. 2, pp. 175–182, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Q. Wu, J. N. Zhao, J. X. Yin, and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing, Singapore, 2001.
View at: Publisher Site | MathSciNet

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

370

Downloads

728

Citations