Positive global solutions of nonlocal boundary value problems for the nonlinear convection reaction-diffusion equations
- Tianfu Ma^{1}Email author and
- Baoqiang Yan^{1}
Received: 21 October 2016
Accepted: 30 December 2016
Published: 10 January 2017
Abstract
In this paper, the nonlocal boundary value problems for a class of nonlinear functional convection reaction-diffusion equations with the singular reaction function are studied by using the method of upper and lower solutions and monotone iterative technique. Some of sufficient results on the existence and uniqueness of positive global solutions or positive solutions for the boundary value problems are presented, which are a generalization of some recent results in the area.
Keywords
MSC
1 Introduction
Hypothesis (\(\mathcal{H}\))
(ii) \(f(x,t,0,0)\ge0\) and there exists a constant \(m_{0}>0\) such that \(f(x,t,u,v)\) is a \(C^{1}\)-function in \((u,v)\) and \(f_{v}(x,t,u,v)\geq 0\) for \(u,v\in[0,m_{0})\).
The purpose of this study is to establish the existence and uniqueness of the positive global solutions or positive solutions for problems (1.1) or problem (1.3). This paper is organized as follows. In Section 2, the discussion focuses on the positive solutions of nonlocal nonlinear functional elliptic boundary value problems (1.3), we first present the maximal and minimal solutions and \(C^{2+\alpha}\) nonnegative solutions by monotone iterative technique and Schauder estimates; lastly, some results on a positive local solution and the uniqueness of positive solutions for problem (1.3) are derived. In Section 3, the discussion focuses on the positive global solutions for nonlocal nonlinear convection reaction-diffusion boundary value problems (1.1), we present some results on the unique fixed solution, a strong solution for problem (1.1) by the means of Collatz monotone operator, and we show that every smooth upper solution of the elliptic problem (3.4) gives rise to a nonincreasing solution of the nonlocal convection reaction-diffusion problem (3.5) and \(u_{t}\le0\) in Ω provided Hypothesis () holds; lastly, the sufficient and necessary conditions of positive global solutions and the uniqueness of positive global solutions for problem (1.1) are both given.
2 Positive solutions of nonlocal nonlinear functional elliptic boundary value problems
It is well known that various assumptions in the previous literature have been made on the reaction term \(f(x,t,u,K*u)\) (we have \(K*u=0\), t, or \((x,t)\)) such as monotonicity, positivity, convexity, concavity, or boundedness, etc., but these assumptions can be relaxed considerably (if not altogether) by using the iteration scheme (cf. [10, 24, 26]). One of the contributions in this paper, of course, in this section will be to emphasize the importance of the applications of upper and lower solutions (cf. [15, 16, 21, 24]), which are defined by the following.
Definition 2.1
Theorem 2.1
Let Hypothesis () hold, and let ǔ, û be a pair of ordered upper and lower solutions of (1.3). If \(f(x,u,K*u)\) is a smooth function on \(\min\hat{u}\le u\le\max\check{u}\). Then there exist two nonnegative solutions ū and \(\underline{u}\) of the problem (1.3) such that \(\hat{u}\leq\underline{u}\leq\bar{u}\leq\check{u}\).
Proof
Corollary 2.1
If solutions \(\{\bar{u}_{\max}\}\) and \(\{\underline{u}_{\min}\}\) are constructed in the proof of Theorem 2.1. Then, for any solution w of the problem (1.3), which satisfies \(\hat{u}\leq w\leq\check{u}\), we have \(\underline{u}_{\min}\leq w\leq\bar{u}_{\max}\).
Proof
In view of the proof of Theorem 2.1, we have \(w=Tw\), \(\bar{u}_{1}=T\check{u}\); since \(w\leq\check{u}\), \(Tw< T\check{u}\), or \(w<\bar{u}_{1}\). By induction, \(w\leq\bar{u}^{(k)}\) for all k, hence \(w\leq\bar{u}_{\max}\). Similarly, \(w\geq\underline{u}_{\min}\), so \(\underline{u}_{\min}\leq w\leq\bar{u}_{\max}\). □
Hypothesis () implies that \(\hat{u}=0\) is a lower solution of (1.3) for domain Ω. In order to find a positive solution, we thus only to find a positive upper solution. To do this, we have a result which is similar to [21] as follows.
Theorem 2.2
Let Hypothesis () hold. Then the problem (1.3) has at least one positive local solution \(u^{+}(x)\).
Proof
As is well known, the monotone iterative scheme for elliptic boundary value problems is based on a positivity lemma which plays a fundamental role in nonlinear elliptic boundary value problems. A lemma (cf. [15]) under consideration is introduced here for the sake of discussing the uniqueness of the positive solutions.
Lemma 2.1
Theorem 2.3
Let β be a function which not identically zero, and let \(\check{u}(x)\), \(\hat{u}(x)\) be a pair of ordered nonnegative upper and lower solutions of (1.3). If the function \(f(x,u,K*u)\) satisfies (2.4), then the positive solution of the problem (1.3) in \(\langle\hat{u},\check{u}\rangle\) is unique.
Proof
3 Positive global solutions of nonlocal functional reaction-diffusion boundary value problems
Definition 3.1
Theorem 3.1
The following corollary is immediate from Theorem 3.1, if g is time independent.
Corollary 3.1
Now if \(u(x)\) is an upper solution of the elliptic problem (1.3), then as we have seen, it can be made the starting point of a monotone decreasing sequence of iterates and we may obtain the corresponding construction solution \(u(x,t)\) which is monotone decreasing on time t. Thus we have the following result.
Theorem 3.2
Proof
Remark 3.1
Theorem 3.2 illustrates that every smooth upper solution \(\bar{u}(x)\) of the elliptic problem (3.4) gives rise to a nonincreasing solution \(u(x,t)\) of the convection reaction-diffusion problem (3.5), and \(u_{t}\leq0\) in Ω provided Hypothesis () holds.
It is well known that the maximum principle of parabolic or elliptic boundary value problems in the method of upper and lower solutions of convection reaction-diffusion boundary value problems plays a fundamental role, especially in the construction of monotone sequences. This role is reflected in Lemma 3.1 which is called the positive lemma (see [15]), for the time-dependent and the steady-state problem, respectively.
Lemma 3.1
Theorem 3.3
Proof
Remark 3.2
We see, from the proof of Theorem 3.3, that the condition (3.10) shows that the pair \(c_{1}\), \(c_{2}\) is a pair of positive upper and lower solutions. So, as a result, Theorem 3.3 may be given in another form as follows.
Corollary 3.2
Clearly, in this situation \(\hat{u}=0\) is a lower solution of the problem (1.1). An immediate consequence from Theorem 3.3 is the following sufficient and necessary conditions for the existence of positive solutions.
Theorem 3.4
Let Hypothesis () hold, and let condition (3.9) hold and not all the three functions are identically zero. If \(f(x,t,u,K*u)\) is a \(C^{1}\)-function on u, \(K*u\in\mathbb {R}^{+}\). Then problem (1.1) has a unique positive solution if and only if there exists a positive upper solution.
We are now in a position to give the uniqueness result of positive global solution for problem (1.1) as follows.
Theorem 3.5
Proof
Remark 3.3
The condition (3.11) in Theorem 3.5 ensures the existence of a unique positive global solution in \(\Omega\times \mathbb {R}^{+}\) but is not necessarily uniformly bounded. As for the discussion of bounded positive global solutions of the problem (1.1) will be still a very interesting work.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers and editors for their valuable comments and suggestions which improve the presentation of the paper.
This work is supported by the NSFC of China (61603226) and the Fund of Science and Technology Plan of Shandong Province (2014GGH201010).
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.
Authors’ Affiliations
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