On Solutions of a Parabolic Equation with Nonstandard Growth Condition

Zhan, Huashui

doi:https://doi.org/10.1155/2020/9397620

Journal of Function Spaces

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

Research Article | Open Access

Volume 2020 | Article ID 9397620 | https://doi.org/10.1155/2020/9397620

On Solutions of a Parabolic Equation with Nonstandard Growth Condition

Huashui Zhan¹

Academic Editor: Maria Alessandra Ragusa

Received03 Nov 2019

Revised01 May 2020

Accepted11 May 2020

Published02 Jun 2020

Abstract

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient on the time variable , and the partial boundary value condition based on a submanifold of . How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

1. Introduction

In this paper, we are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition where is a function, is a function, , , and is a bounded domain with a smooth boundary . These kinds of equations can be regarded as nonlinear variation of the classical heat equation, the well-known paradigm to explain the diffusion process. In particular, when , is a constant; equation (1) is with the type which includes the so-called porous medium equation (PME), the fast diffusion equation (FDE) when , and the slow diffusion equation (SDE) when Correspondingly, if , then equation (1) is a degenerate parabolic equation and has a slow diffusion characteristic. If , then equation (1) is a singular parabolic equation and has a fast diffusion characteristic.

In fact, equation (2) itself has a wide number of applications, ranging from plasma physics to filtration in porous media, thin films, Riemannian geometry, relativistic physics, and many others. It has at the same time tested as the testing ground from the development of new methods of analytical investigation, since it offers a variety of surprising phenomena that strongly deviate from the heat equation standard, for example, free boundary, limited regularity, mass loss, and extinction or quenching. There exists an abundant literature on these topics, one can refer to [1–10] and the references therein. In this literature, the nonlinearities are with power types that lead to degenerate or singular parabolicity, and we can gather these collectively under the name “the porous medium equation with standard growth conditions.” Since, in equation (1), the nonlinearity is with variable exponent, so it is called as “with nonstandard growth condition.”

From another perspective, since , equation (1) is a special case of the usual reaction-diffusion equation

If is degenerate in the interior of , this equation is a hyperbolic-parabolic mixed type equation theoretically. To ensure the uniqueness of a weak solution to this equation, the entropy condition and the corresponding entropy solution are required, one can refer to references [11–16] for the details. In this paper, we assume that

As a consequence, equation (1) is a pure nonlinear parabolic equation and has not the hyperbolic characteristic.

The porous medium equation with nonstandard growth conditions was first proposed by Antontsev and Shmarev in [17]. The existence, uniqueness, and localization properties of solutions to equation had been studied. Another property proved in [17] is the finite speed of propagation, which permits one to consider the free boundary problem. The work [18] by Duque etc. implemented a finite element method with adaptive mesh to approximate the solution of the porous medium equation with nonstandard growth conditions in 2D domains with free boundary. The equidistribution principle deduced by de Boor [19] and the moving mesh for partial differential equations by Huang and Russell [20] were used there. Recently, when with , and the well-posedness of weak solutions to equation (1) has been studied by the author [21]. Roughly speaking, if satisfies (6) and the uniqueness of weak solution to equation (1) with the initial value has been proved in [21]. This result can be comprehended as that, the degeneracy of diffusion coefficient on the boundary (6) acts as a role as the usual boundary value condition

In this paper, different from [21], the diffusion coefficient is dependent on the time variable ; moreover, we merely assume that satisfies (4) and do not restrict the similar condition as (7). As a compensation, a partial boundary value condition is imposed as follows. One can see that, unlike the usual Dirichlet boundary value condition (9), in which , is a cylinder, appearing in (10) is just a submanifold and generally can not be expressed as a cylinder type as with . The first aim of this paper is to study the stability of weak solutions based on this partial boundary value condition. Another aim of this paper is to study the existence of a strong solution to equation (1). The details are provided below. It is well-known that, for the usual porous medium equation there holds and is with the Hölder continuity [5]. However, for a porous medium equation with nonstandard growth conditions, weak solutions in [17, 18, 21–30] do not satisfy the regularity as (12). Thus, in this paper, we will try to make up for some gaps in such a field, considering that the weak solutions of equation (1) have the property (12).

Definition 1. Let be a nonnegative function satisfying If for any function , there is then is said to be a weak solution of the initial-boundary value problem of equation (1), provided that initial value (6) is true in the sense and the partial boundary value condition (10) is true in the sense of the trace.

Definition 2. Let be a nonnegative function satisfying (14) and then is said to be a strong solution of the initial-boundary value problem of equation (1), provided that initial value (6) is true in the sense of (15), and the partial boundary value condition (10) is true in the sense of the trace.
Since satisfies (4), (16) means that and exist almost everywhere in . This is the reason that we call is a strong solution of equation (1).
From Definition 2, for all , we have which implies that

Thus, if is a strong solution of equation (1), then it is a weak solution.

Theorem 3. If is a function, , , , satisfies satisfies (4), is a continuous function on and when then the initial boundary value problem of equation (1) has a nonnegative weak solution.

Theorem 4. If is a function, , , , satisfies (19), satisfies (4) and for any , , is a continuous function satisfies (20), then the initial boundary value problem of equation (1) has a nonnegative strong solution.
Throughout this paper, the constant may be different from one to another, represents the constant depends on , and represents the constant depends on .

Theorem 5. Let be a weak solution of equation (1) with the initial value (8) and with the partial homogenous value condition (10), If 0 , is a Lipschitz function on , for every given , is a differential function on the spatial variables , , and satisfies then the solution is unique.
One can see that there are functions satisfying condition (24). For example, if is a relatively open set of , and let . Then, for any , is a function satisfying (24).

We note that, if is a constant, then condition (22) is naturally true. By the way, just as one reviewer has suggested, one can study Theorem 3 and Theorem 4 when and is just a Carathodory function.

2. The Proof of Theorems 3-4

In this section, we give the proof of the existence theorems.

Proof of Theorem 3. By the monotone convergent method [5], since when , there is a nonnegative weak solution of the following initial-boundary value problem which satisfies Multiplying (26) by , then we obtain In the first place, by a similar calculation as that in [21], we can extrapolate that and show that weakly in .
In the second place, for all , we have from this equality, by (27) and (31), it is easy to get Moreover, from (27), there is a nonnegative function, weakly star in . For any , by (34), we can show that Since when , then . Accordingly, In addition, we have where . Thus, Since , Aubin’s compactness theorem in [31] yields that there is a function such that strongly in . Then, almost everywhere in , by that weakly star in , we know that . The arbitraries of yields almost everywhere in .
Thus, since and are continuous functions on ), we have Let in (29). By (32), (39), and (40), we know satisfies (13) and (14).
The last but not the least, by a process of limit, in a similar way as that in [17], we can choose the test function in which and is the characteristic function of . Then Let and . Then we have (15).
Finally, since when in (10), by (32), we know that the trace of on is well defined, the details are given in Section 4 as follows. Till now, we have shown that is a weak solution of equation (1) with the initial value (6) and the partial boundary value condition (10).

Proof of Theorem 4. We consider the regularized problem (26) and obtain (27)–(32) as in the proof Theorem 3.
Let us multiply on both sides of equation (26), integrate it over . Then Since , we have Since , by the assumption , we have By the assumption and when , we have Moreover, we have Now, combining with (44)–(51), we have Then Since when , by Sobolev embedding theorem, (32) and (53) implies that a.e. in . Since weakly star in , by the uniqueness property of the weak convergence, we know that .
Accordingly, , we have as well as by that is a contunuous function.
Moreover, for any , Thus, Let in (29). By (32), (35), (39), (56), and (57), we know satisfies (14) and (16). The initial value (6) and the partial boundary value condition have the same meaning as those in Theorem 3.

3. The Stability Based on the Partial Boundary Value Condition

In this section, we will prove Theorem 5.

Lemma 6. Let for any , . For any continuous function , , ,

This Lemma can be seen as a generalization of Lemma 2.2 in [32]. We postpone the reader to it for details.

Theorem 7. Let , be a Lipschitz function on , with that if and for every given , be a differential function on the spatial variables , and satisfy If are two nonnegative weak solutions of equation (1) with the initial value , respectively, with the same homogeneous partial boundary value condition and satisfying then

Proof. Let , . Here and the after, is a positive integer. Supposed that is the characteristic function of , since and are with the same homogeneous partial boundary value condition (61), we then choose as the test function. Thus In the first place In the second place, by the Hölder inequality, using (59) and (64), we obtain In the third place, for simplism, let us denote that as a consequence Since condition (62) implies using the Lebesgue dominated convergence theorem, we have which yields In the fourth place, since , using (64) and the Lebesgue dominated convergence theorem, is obviously.
In the fifth place, since satisfies (60), using the Lebesgue dominated convergence theorem, we can extrapolate Also by the Hölder inequality, we are easily to obtain In the sixth place, At last, by Lemma 6, choosing , by (64), we have Then, letting in (66), we have By the Gronwall inequality, we have By the arbitraries of , we have At last, since conditions (22), (23), and (24) imply all the assumptions in Theorem 7 are true, we have Theorem 4.

4. The Explanation of the Partial Boundary Value Condition

In [21], condition (7) guarantees

then we can define the trace of on , and the homogeneous boundary value condition (9) is valid. But in this paper, we do not assume

which is similar as condition (7), how to guarantee the partial boundary value condition (10) is true in the sense of the trace? Actually, by the character of defined in Definition 1it is not difficult to show that

Now, for a given small constant and for any , if we denote where is the distance function from the boundary , then

If we denote that then

Since satisfies (4) and (86), we have

Then, we can define the trace of on . By the arbitrariness of , we can define the trace of on . In other words, no matter whether condition (84) is true or not, we can always define the trace of (also ) on due to that on , and so the partial homogenous value condition (10) is always valid. Certainly, we should point out that, on the remained part of the boundary whether the trace of can be defined or not is unknown, unless condition (84) is supplied.

5. Conclusion

Since the beginning of this century, the evolutionary -Laplacian equations have been studied by many mathematicians in [32–37]. So far, however, there has been little discussion about the porous medium equation with nonstandard growth conditions. Therefore, this study makes a major contribution to research in the related fields by imposing a reasonable partial boundary value condition matching up with the equation. The unusual is that this partial boundary value condition is based on a submanifold . In addition, since the equation is with nonstandard growth conditions, how to deal with the nonlinearity becomes difficult, especially when we try to prove that .

Data Availability

There is not any data in this paper.

Conflicts of Interest

The author declares that he has no competing interests.

Acknowledgments

The paper is supported by Natural Science Foundation of Fujian Province (2019J01858), supported by the Science Foundation of Xiamen University of Technology, China.

References

J. L. Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Equations of Porous Medium Type, Oxford University Press, 2006.
View at: Publisher Site
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, USA, 1964.
J. Smoller, Shock Waves and Reaction Diffusion Equations, Academic Press, New York, NY, USA, 1983.
View at: Publisher Site
A. S. Alexander, A. G. Victor, P. K. Sergei, and P. M. Alexander, Blow-Up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, NY, USA, 1995.
Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations, Word Scientific Publishing, Singapore, 2001.
View at: Publisher Site
E. DiBenedetto, Degenerate Parabolic Equations, Spring-Verlag, New York, NY, USA, 1993.
View at: Publisher Site
A. Razani, “Subsonic detonation waves in porous media,” Physica Scripta, vol. 94, no. 8, article 085209, 2019.
View at: Publisher Site | Google Scholar
A. Razani, “Chapman-Jouguet travelling wave for a two-steps reaction scheme,” Italian Journal of Pure and Applied Mathematics, vol. 41, pp. 544–553, 2018.
View at: Google Scholar
S. Benbernou, S. Gala, and M. A. Ragusa, “On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space,” Mathematical Methods in the Applied Sciences, vol. 37, no. 15, pp. 2320–2325, 2014.
View at: Publisher Site | Google Scholar
S. Gala and M. A. Ragusa, “Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices,” Applicable Analysis, vol. 95, no. 6, pp. 1271–1279, 2016.
View at: Publisher Site | Google Scholar
Y. Li and Q. Wang, “Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic–hyperbolic equations,” Journal of Differential Equations, vol. 252, no. 9, pp. 4719–4741, 2012.
View at: Publisher Site | Google Scholar
B. Andreianov, M. Bendahmane, K. H. Karlsen, and S. Ouaro, “Well-posedness results for triply nonlinear degenerate parabolic equations,” Journal of Differential Equations, vol. 247, no. 1, pp. 277–302, 2009.
View at: Publisher Site | Google Scholar
K. Kobayasi and H. Ohwa, “Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle,” Journal of Differential Equations, vol. 252, no. 1, pp. 137–167, 2012.
View at: Publisher Site | Google Scholar
H. Zhan, “The solutions of a hyperbolic–parabolic mixed type equation on half-space domain,” Journal of Differential Equations, vol. 259, no. 4, pp. 1449–1481, 2015.
View at: Publisher Site | Google Scholar
H. Zhan and Z. Feng, “Stability of hyperbolic-parabolic mixed type equations with partial boundary condition,” Journal of Differential Equations, vol. 264, no. 12, pp. 7384–7411, 2018.
View at: Publisher Site | Google Scholar
H. Zhan and Z. Feng, “Partial boundary value condition for a nonlinear degenerate parabolic equation,” Journal of Differential Equations, vol. 267, no. 5, pp. 2874–2890, 2019.
View at: Publisher Site | Google Scholar
S. N. Antontsev and S. I. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 3, pp. 515–545, 2005.
View at: Publisher Site | Google Scholar
J. C. M. Duque, R. M. P. Almeida, and S. N. Antontsev, “Numerical study of the porous medium equation with absorption, variable exponents of nonlinearity and free boundary,” Applied Mathematics and Computation, vol. 235, pp. 137–147, 2014.
View at: Publisher Site | Google Scholar
C. de Boor, “Good approximation by splines with variable knots. II,” in Conference on the Numerical Solution of Differential Equations, G. A. Watson, Ed., vol. 363 of Lecture Notes in Mathematics, pp. 12–20, Springer, Berlin, 1974.
View at: Publisher Site | Google Scholar
W. Huang and R. D. Russell, “Moving mesh strategy based on a gradient flow equation for two-dimensional problems,” SIAM Journal on Scientific Computing, vol. 20, pp. 998–1015, 1998.
View at: Publisher Site | Google Scholar
H. Zhan, “On the solutions of a porous medium equation with exponent variable,” Discrete Dynamics in Nature and Society, vol. 2019, Article ID 9290582, 15 pages, 2019.
View at: Publisher Site | Google Scholar
L. Diening, P. Nägele, and M. Růžička, “Monotone operator theory for unsteady problems in variable exponent spaces,” Complex Variables and Elliptic Equations, vol. 57, no. 11, pp. 1209–1231, 2010.
View at: Publisher Site | Google Scholar
S. Antontsev and S. Shmarev, “Anisotropic parabolic equations with variable nonlinearity,” Publicacions Matemàtiques, vol. 53, pp. 355–399, 2009.
View at: Publisher Site | Google Scholar
A. H. Erhardt, “Compact embedding for p(x,t)-Sobolev spaces and existence theory to parabolic equations with p(x,t)-growth,” Revista Matemática Complutense, vol. 30, no. 1, pp. 35–61, 2017.
View at: Publisher Site | Google Scholar
S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations, Atlantis Press, Paris, 2015.
View at: Publisher Site
S. Antontsev and S. Shmarev, “Vanishing solutions of anisotropic parabolic equations with variable nonlinearity,” Journal of Mathematical Analysis and Applications, vol. 361, no. 2, pp. 371–391, 2010.
View at: Publisher Site | Google Scholar
S. Antontsev and S. Shmarev, “On the localization of solutions of doubly nonlinear parabolic equations with nonstandard growth in filtration theory,” Applicable Analysis, vol. 95, no. 10, pp. 2162–2180, 2015.
View at: Publisher Site | Google Scholar
B. Amaziane, L. Pankratov, and A. Piatnitski, “Nonlinear flow through double porosity media in variable exponent Sobolev spaces,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2521–2530, 2009.
View at: Publisher Site | Google Scholar
R. M. P. Almeida, S. N. Antontsev, and J. C. M. Duque, “Discrete solutions for the porous medium equation with absorption and variable exponents,” Mathematics and Computers in Simulation, vol. 137, pp. 109–129, 2017.
View at: Publisher Site | Google Scholar
S. Lian, W. Gao, C. Cao, and H. Yuan, “Study of the solutions to a model porous medium equation with variable exponent of nonlinearity,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 27–38, 2008.
View at: Publisher Site | Google Scholar
J. P. Aubin, “Un théoràme de compacité,” Comptes Rendus de l'Académie des Sciences, vol. 256, pp. 5042–5044, 1963.
View at: Google Scholar
Z. Li, B. Yan, and W. Gao, “Existence of solutions to a parabolic p(x)-Laplace equation with convection term via L^∞ estimates,” Electronic Journal of Differential Equations, vol. 46, pp. 1–21, 2015.
View at: Google Scholar
S. Antontsev and S. Shmarev, “Parabolic equations with double variable nonlinearities,” Mathematics and Computers in Simulation, vol. 81, no. 10, pp. 2018–2032, 2011.
View at: Publisher Site | Google Scholar
S. Lian, W. Gao, H. Yuan, and C. Cao, “Existence of solutions to an initial Dirichlet problem of evolutional p(x)-Laplace equations,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 29, no. 3, pp. 377–399, 2012.
View at: Publisher Site | Google Scholar
J. Aramaki, “Hölder continuity with exponent (1 + α)/2 in the time variable for solutions of parabolic equations,” Electronic Journal of Differential Equations, vol. 96, pp. 1–6, 2015.
View at: Google Scholar
A. S. Tersenov and A. S. Tersenov, “Existence of Lipschitz continuous solutions to the Cauchy–Dirichlet problem for anisotropic parabolic equations,” Journal of Functional Analysis, vol. 272, no. 10, pp. 3965–3986, 2017.
View at: Publisher Site | Google Scholar
C. Cowan and A. Razani, “Singular solutions of a p−Laplace equation involving the gradient,” Journal of Differential Equations, vol. 269, no. 4, pp. 3914–3942, 2020.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Huashui Zhan. 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.

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