Abstract
The anisotropic parabolic equations with variable exponents are considered. If some of diffusion coefficients \(\{b_{i}(x)\}\) are degenerate on the boundary, the others are always positive, then how to impose a suitable boundary value condition is researched. The existence of weak solutions is proved by the parabolically regularized method. The stability of weak solutions, based on the partial boundary value condition, is established by choosing a suitable test function.
Similar content being viewed by others
1 Introduction and the main results
Recently, the anisotropic parabolic equations with the variable exponents
were studied by Antontsev and Shmarev [1], Tersenov [2, 3], and some essential characteristics different from the evolutionary p-Laplacian equations were revealed. Zhan [4, 5] studied the equations
and showed some essential characteristics different from equation (1.1). Here,
In this paper, we study the equation
with the initial value condition
and with a partial boundary value condition
where \(\Sigma_{1}\subseteq\partial\Omega\) is a relatively open subset. A similar partial boundary value condition was imposed on the equation
and a new approach to prescribe the boundary value condition rather than define the Fichera function was formulated by Yin and Wang [6]. However, since equation (1.4) is anisotropic and with the variable exponents, the method of [6] seems difficult to be applied to equation (1.4). In what follows, we will try to depict \(\Sigma_{1}\) in another way. Moreover, instead of depicting the explicit formula of \(\Sigma _{1}\), we will try to find the other conditions to substitute the boundary value condition.
Instead of condition (1.3), we assume that \(x\in\Omega\), \(b_{i}(x)>0\), and
Here, \(\{i_{1}, i_{2}, \ldots, i_{k}\}\cup\{j_{1}, j_{2}, \ldots, j_{l}\}=\{ 1, 2, \ldots, N\}\), \(k+l=N\). For the sake of simplicity, we denote that
and assume that \(p_{0}>1\).
Let us introduce the basic definition and the main results. First of all, for any small constant \(\eta>0\), we define
Conditions (1.8)–(1.9) assure that this set is an open subset of Ω.
Definition 1.1
If a function \(v(x,t)\) satisfies
and for \(\varphi\in L^{2}(0,T; W^{1,p^{0}}(\Omega)), \varphi |_{x\in\partial\Omega}=0\),
then we say that \(v(x,t)\) is a weak solution of equation (1.1) with initial value condition (1.5), provided that
Besides, if the partial boundary value condition (1.6) is satisfied in the sense of the trace, then we say that \(v(x,t)\) is a weak solution of the initial-boundary value problem (1.4)–(1.6).
Here and in what follows, \(p'=\frac{p}{p-1}\) as usual.
Theorem 1.2
If \(p_{0}>1\), \(b_{i}(x)\) satisfies conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\), \(a(s)\) is a continuous function,
then equation (1.4) with initial value (1.5) has a solution. If
then there exists a solution of the initial-boundary value problem (1.4)–(1.6).
Theorem 1.3
Let \(p_{0}>1\), \(b_{i}(x)\) satisfy conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\), \(a(s)\) be a Lipschitz function and for every \(1\leq r\leq l\), \(\int_{\Omega}b_{j_{r}}^{-\frac {1}{p_{j_{r}}-1}}(x)\,dx<\infty\). If \(v(x,t)\) and \(u(x,t)\) are two solutions of equation (1.4),
then
Theorem 1.4
If \(p_{0}>1\), \(b_{i}(x)\) satisfies conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\), \(a(s)\) is a Lipschitz function. Let \(v(x,t)\) and \(u(x,t)\) be two solutions of equation (1.4). If
and for every \(1\leq r\leq l\),
then the stability (1.15) is true. Here,
If \(a(s)\equiv0\) in equation (1.4), the similar conclusion as Theorem 1.4 was obtained in [5], where the partial boundary \(\Sigma_{1}\) was depicted as follows:
In fact, letting φ be a nonnegative \(C^{1}\) function, satisfying
the partial boundary \(\Sigma_{1}\) can be depicted by φ as
By this token, the exact partial boundary \(\Sigma_{1}\), such that the partial boundary value condition (1.6) matches up the nonlinear degenerate parabolic equation, should satisfy that
and we can depict it as
for any φ satisfying (1.20). However, if we really choose \(\Sigma_{1}\) as (1.22), it lacks the technical support to obtain the stability of the weak solutions for the time being. Anyway, by adopting some ideas and techniques in [4, 5], in some special cases, we can prove the stability of the weak solutions independent of the boundary value condition.
Theorem 1.5
If \(p_{0}>1\), \(b_{i}(x)\) satisfies conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\), \(a(s)\) is a Lipschitz function. Let \(v(x,t)\) and \(u(x,t)\) be two solutions of equation (1.4) only with the initial values \(v_{0}(x)\) and \(u_{0}(x)\), respectively, but without any boundary value condition. If condition (1.17) is true, and
for every \(1\leq r\leq k\), then the stability (1.15) is true.
One can see that no boundary value condition is required in Theorem 1.5. From my own perspective, condition (1.23) is an alternative of the partial boundary value condition (1.6). By the way, for the following reaction–diffusion equation
with
we had conjectured that a partial boundary value condition should be imposed. This conjecture was partially proved in [7, 8].
2 The proof of existence
By a similar method as in [4], we can prove the following.
Lemma 2.1
If \(\int_{\Omega}b_{i}^{-\frac {1}{p_{i}-1}}(x)\,dx<\infty\), \(v(x,t)\) is a weak solution of equation (1.4) with initial condition (1.5). Then, for any given \(t\in[0,T)\),
and the trace of v on the boundary ∂Ω can be defined in the traditional way.
We omit the details of the proof here. By this lemma, we know that if \(b_{i}(x)\) satisfies (1.8), (1.9) and if for every \(1\leq r\leq l\), \(\int _{\Omega}b_{j_{r}}^{-\frac{1}{p_{j_{r}}-1}}(x)\,dx<\infty\), then (2.1) is satisfied. Thus, we can define the trace of v on the boundary ∂Ω.
Consider the regularized equation
with the initial-boundary condition
Here, \(v_{0\varepsilon}(x)\in C_{0}^{\infty}(\Omega)\) and is strongly convergent to \(v_{0}(x)\) in \(W_{0}^{1,p^{0}}(\Omega)\).
Then, by Wu [9], we know that problem (2.2)–(2.4) has a unique solution \(v_{\varepsilon}\in L^{\infty}(Q_{T})\), \(v_{\varepsilon}\in W^{1,p_{0}}_{0}(\Omega)\).
Proof of Theorem 1.2
Multiplying (2.2) by \(v_{\varepsilon}\) and integrating it over \(Q_{T}\) yield
then
and
Hence, by (2.5), (2.6), and (2.7), there exists a function v and an n-dimensional vector \(\overrightarrow{\xi}= ({\xi_{1}}, \ldots,{\xi _{n}})\) such that
and \(v_{\varepsilon}\rightarrow v\) \(a.e.\in Q_{T}\),
Here, \(r<\frac{Np_{0}}{N-p^{0}}\).
Now, similar to [1], we can show that
and by Wu [9], by a process of the limit, we are able to prove that
for any function \(\varphi\in L^{2}(0,T; W^{1,p^{0}}(\Omega))\), \(\varphi |_{x\in\partial\Omega}=0\). Thus, \(v(x,t)\) satisfies (1.10) and (1.11). Moreover, according to Lemma 2.1, the partial boundary value condition (1.6) is satisfied in the sense of trace.
Now, we can prove the initial value (1.5) in a similar way as that in [10]. In detail, for small given \(r>0\), denote \(D_{r}=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega)\leq r\}\). For large enough \(m, n\), denoting that \(v_{m}(x,t)=v_{\varepsilon=\frac{1}{m}}(x,t)\), we declare that
where \(c_{r}(t)\) is independent of \(m,n\), and \(\lim_{t\rightarrow 0}c_{r}(t)=0\). In fact, by (2.2), for any \(t\in[0, T)\), we have
For small \(\eta>0\), let
Obviously, \(l_{\eta}(s)\in C(\mathbb{R})\) and
Clearly, if we denote \(A_{\eta}(s)=\int_{0}^{s}L_{\eta}(s)\,ds\),
and
Suppose that \(\xi(x)\in C_{0}^{1}(D_{r})\) such that
and choose \(\varphi=\xi L_{\eta}(v_{m}-v_{n})\) in (2.10), then
Clearly,
and
Noticing that \(\xi\in C_{0}^{1}(\Omega)\), \(a(s)\) is a Lipschitz function, using Hölder’s inequality of the variable exponent Sobolev space, by (2.14), we easily deduce that
At the same time,
Let \(\eta\rightarrow0\). By (2.11)–(2.19), we can obtain
By \(v_{\varepsilon}\in L^{\infty}(Q_{T})\) and the unform estimates (2.6)–(2.7), we know that \(c_{r}(t)\) is independent of \(m,n\).
Now, for any given small r, if \(m,n\) are large enough, by (2.9), we have
By (2.20), similar to the usual evolutionary p-Laplacian equation (Chap. 2, [9]), we have (1.12). Then Theorem 1.2 is proved.
Certainly, the initial value condition (1.5) can be right in the other sense; for example, in [11], it has the form
Also, the existence of weak solutions can be proved in other ways. Here, we would like to mention some recent related papers [12–14]. □
3 The stability of the initial-boundary value problem
Theorem 3.1
If \(p_{0}>1\), \(b_{i}(x)\) satisfies conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\), \(a(s)\) is a Lipschitz function and for every \(1\leq r\leq l\), \(\int_{\Omega}b_{j_{r}}^{-\frac {1}{p_{j_{r}}-1}}(x)\,dx<\infty\), \(g^{i}(x)\) satisfies
If \(v(x,t)\) and \(u(x.t)\) are two solutions of equation (1.4) with the same homogeneous value
and with different initial values \(u_{0}(x)\) and \(v_{0}(x)\), then
Proof
Since \(\int_{\Omega}b_{j_{r}}^{-\frac {1}{p_{j_{r}}-1}}(x)\,dx<\infty\), by Lemma 2.1, we can choose \(\varphi =\chi_{[\tau,s]}L_{\eta}(v - u)\) in (1.11), where \(\chi_{[\tau ,s]}\) is the characteristic function of \([\tau, s]\subset(0, T)\). Then
At first, we have
By Lemma 3.1 from [11], we have
Moreover, since \(g^{i}(x)\) satisfies condition (3.1)
by (3.4), \(a(s)\) is a Lipschitz function, we have
Now, let \(\eta\rightarrow0\) in (3.2). Then
By Gronwall’s inequality, letting \(\tau\rightarrow0\), we have
Theorem 3.1 is proved.
One can see that condition (3.1) is used to prove (3.4). In fact, without this condition, the conclusion of Theorem 3.1 is still true. This is Theorem 1.3. □
Proof of Theorem 1.3
From the above proof of Theorem 3.1, we only need to prove that
without condition (3.1). Let us give an explanation. Noticing
if \(\{\Omega: \vert v-u \vert =0\}\) is a subset of Ω with a positive measure, then
At the same time, if \(\{\Omega: \vert v-u \vert =0\}\) is a subset of Ω with zero measure, since \(b_{i}(x)\) satisfies (1.8)–(1.9), and for every \(1\leq r\leq l\), \(\int_{\Omega}b_{j_{r}}^{-\frac{1}{p_{j_{r}}-1}}(x)\,dx<\infty\), we have
Then
Thus, Theorem 1.3 is true. □
4 The stability based on the partial boundary value condition
Theorem 4.1
If \(p_{0}>1\), \(b_{i}(x)\) satisfies conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\) satisfies (3.5), \(a(s)\) is a Lipschitz function. Let \(v(x,t)\) and \(u(x,t)\) be two solutions of equation (1.4). If the initial values \(u_{0}(x)\) and \(v_{0}(x)\) are different, while the partial boundary values satisfy
then the stability (1.15) is true, provided that for every \(1\leq r\leq l\), condition (1.17) is true, i.e.,
Proof
Let \(\Omega_{\eta}=\{x\in\Omega:\sum_{r=1}^{l}b_{j_{r}}(x)>\eta\}\), and
Then, if \(x\in\Omega\setminus\Omega_{\eta}\), \(\phi_{\eta x_{i}}=\frac{1}{\eta}(\sum_{r=1}^{l} b_{j_{r}}(x))_{x_{i}}\), while \(x\in \Omega_{\eta}\), \(\phi_{x_{i}}=0\).
Let \(\varphi=\chi_{[\tau,s]}\phi_{\eta}L_{\eta}(v-u)\) be the test function in (1.11). Then
At first,
and
Secondly, by Hölder’s inequality of the variable exponent Sobolev space, we have
Here, \(p^{1}_{i_{r}}=p^{+}_{i_{r}}\) or \(p^{-}_{i_{r}}\) according to
or
one can refer to Lemma 2.1 of [4]. \(q_{j_{r}}(x)=\frac {p_{j_{r}}(x)}{p_{j_{r}}(x)-1}\), \(q^{1}_{i_{r}}\) has a similar sense.
Let \(\Sigma_{2}=\partial\Omega\setminus\Sigma_{1}\), and
Then
Since
by the definition of the trace, we have
Moreover, since
we have
According to (4.5)–(4.8), we have
Thirdly, for the last term of the left-hand side of (4.2), we have
By condition (1.17),
Then
Fourthly, since \(g^{i}(x)\) satisfies condition (3.5), we have
At last, we have
Now, let \(\eta\rightarrow0\) in (4.2). By (4.3), (4.4), (4.9), (4.11), (4.12), and (4.13), we have
By Gronwall’s inequality, we have
Let \(\tau\rightarrow0\). Then
□
Proof of Theorem 1.4
If \(\vert g^{i}(x) \vert \leq c\) and \(a(s)\) is a Lipschitz function, for every r satisfies (1.17), similar to the proof of Theorem 1.3 in Sect. 3, combining with Theorem 4.1, we know Theorem 1.4 is true. □
5 The stability without boundary value condition
Theorem 5.1
Let \(v(x,t)\) and \(u(x,t)\) be two solutions of equation (1.4) with different initial values \(v_{0}(x)\) and \(u_{0}(x)\), respectively. If \(p_{0}>1\), \(b_{i}(x)\) satisfies conditions (1.8), (1.9), \(g^{i}(x)\in C^{1}(\overline{\Omega})\) satisfies (3.5), \(a(s)\) is a Lipschitz function, conditions (1.17) and (1.23) are true, then the stability (1.15) is true.
Proof
As in the proof of Theorem 4.1, let \(\varphi=\chi _{[\tau,s]}\phi_{\eta}L_{\eta}(v-u)\) be the test function and obtain (4.2)–(4.4).
Since
by condition (1.23), we have (4.9). At the same time, we have (4.10)–(4.15). The proof is complete. □
Proof of Theorem 1.5
If \(\vert g^{i}(x) \vert \leq c\) and \(a(s)\) is a Lipschitz function, condition (1.23) is true. Similar to the proof of Theorem 1.3 in Sect. 3, combining with Theorem 5.1, we know Theorem 1.5 is true. □
References
Antontsev, S., Shmarev, S.: Existence and uniqueness for doubly nonlinear parabolic equations with nonstandard growth conditions. Differ. Equ. Appl. 4(1), 67–94 (2012)
Tersenov Alkis, S.: The one dimensional parabolic \(p(x)\)-Laplace equation. Nonlinear Differ. Equ. Appl. 23, 27 (2016). https://doi.org/10.1007/s00030-016-0377-y
Tersenov Alkis, S., Tersenov Aris, S.: Existence of Lipschitz continuous solutions to the Cauchy–Dirichlet problem for anisotropic parabolic equations. J. Funct. Anal. 272, 3965–3986 (2017)
Zhan, H.: The stability of the anisotropic parabolic equation with the variable exponent. Bound. Value Probl. 2017, 134 (2017). https://doi.org/10.1186/s13661-017-0868-8
Zhan, H.: The well-posedness of an anisotropic parabolic equation based on the partial boundary value condition. Bound. Value Probl. 2017, 166 (2017). https://doi.org/10.1186/s13661-017-0899-1
Yin, J., Wang, C.: Evolutionary weighted p-Laplacian with boundary degeneracy. J. Differ. Equ. 237, 421–445 (2007)
Zhan, H.: On a hyperbolic-parabolic mixed type equation. Discrete Contin. Dyn. Syst., Ser. S 10(3), 605–624 (2017)
Zhan, H.: The solutions of a hyperbolic-parabolic mixed type equation on half-space domain. J. Differ. Equ. 259, 1449–1481 (2015)
Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations. Word Scientific, Singapore (2001)
Zhan, H.: The solution of convection-diffusion equation. Chin. Ann. Math. 34(2), 235–256 (2013) (in Chinese)
Antontsev, S.V., Shmarev, S.: Parabolic equations with double variable nonlinearities. Math. Comput. Simul. 81, 2018–2032 (2011)
Alaoui, M.K., Messaoudi, S.A., Khenous, H.B.: A blow-up result for nonlinear generalized heat equation. Comput. Math. Appl. 68(12), 1723–1732 (2014)
Al-Smail, J.H., Messaoudi, S.A., Talahmeh, A.A.: Well-posedness and numerical study for solutions of a parabolic equation with variable-exponent nonlinearities. Int. J. Differ. Equ. 2018, Article ID 9754567 (2018)
Messaoudi, S.A., Talahmeh, A.A., Al-Smail, J.H.: Nonlinear damped wave equation: existence and blow-up. Comput. Math. Appl. 74, 3024–3041 (2017)
Acknowledgements
The author would like to thank SpringerOpen Accounts Team for kindly agreeing to give me a discount of the paper charge if my paper can be accepted.
Availability of data and materials
Not applicable.
Funding
The paper is supported by the Natural Science Foundation of Fujian province (no: 2015J01592), supported by the Science Foundation of Xiamen University of Technology, China.
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The author declares that he has no competing interests.
Additional information
Abbreviations
Not applicable
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhan, H. On stability with respect to boundary conditions for anisotropic parabolic equations with variable exponents. Bound Value Probl 2018, 27 (2018). https://doi.org/10.1186/s13661-018-0947-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-018-0947-5