Periodic wave solutions of a non-Newtonian filtration equation with an indefinite singularity

Abstract: This paper is concerned with the existence of periodic wave solutions for a type of nonNewtonian filtration equations with an indefinite singularity. A sufficient criterion for the existence of periodic wave solutions for non-Newtonian filtration equation is provided via an innovative method of combining a new continuation theorem with coincidence degree theory as well as mathematical analysis skills. The novelty of the present paper is that it is the first time to discuss the existence of periodic wave solutions for the indefinite singular non-Newtonian filtration equations. Finally, two numerical examples are presented to illustrate the effectiveness and feasibility of the proposed criterion in the present paper.


Introduction
In this paper, we consider the periodic wave solutions problem for a type of non-Newtonian filtration equation with an indefinite singularity as follows: where p > 1, m > 0, f ∈ C(R, R), h ∈ C(R × R, R). In this equation, the function 1 y m may have a singularity at y = 0. Besides this, the signs of h(t, x) are all allowed to change.
In recent years, the solitary wave and periodic wave solutions for the non-Newtonian filtration equation have been received great attention. In 2014, Lian etc. [8] [11][12][13].
In Eq (1.1), the signs of function h are allowed to change which means that the singularity of 1 y m has a singularity at y = 0 can be classified neither as repulsive type nor as attractive type. In this paper, we will use the theorem belonging to [14] to obtain the existence of periodic wave solutions for Eq (1.1). To the best of our knowledge, there is no paper to use the theorem in [14] for studying the non-Newtonian filtration equations with an indefinite singularity, the main purpose is to recommend a new method for the research of non-Newtonian filtration equations with an indefinite singularity. Recent years, second-order indefinite singular equations have been studied by some researchers. Hakl and Zamora [15] studied a second-order indefinite singular equations by using Leray-Schauder degree theory. Fonda and Sfecci [16] investigated the periodic problem of Ambrosetti-Prodi type having a nonlinearity with possibly one or two singularities. In the present paper, we will generalize secondorder indefinite singular equations to Eq (1.1). Hence, our research can enrich and develop the study of second-order singular equations. The topics of solitary wave solutions, periodic wave, and traveling wave solutions are interesting. Recently, there are many superior works on these topics, see them in [17][18][19][20][21][22][23][24][25][26][27]. For Eq (1.1), assume that there is a continuous function h(s) such that h(t, x) = −h(x + ct) = −h(s), where c ∈ R. Let y(t, x) = u(s) with s = x + ct be the solution of Eq (1.1), then Eq (1.1) is changed into the following equation: where φ p (u) = |u| p−2 u, p > 1, m > 0, f, h ∈ C(R, R).  The highlights of this paper are threefold: (1) In this paper, we studied a new non-Newtonian filtration equation with an indefinite singularity which is different from the existing non-Newtonian filtration equations, see e.g., [3,[8][9][10][11][12].
(2) We creatively use a new continuation theorem to study a class of strongly nonlinear equations. For estimating the prior bounds of periodic wave solutions, we develop some inequality methods and mathematical analysis skills.
(3) Different from the previous results, we introduce a new unified framework to deal with the existence of periodic wave solutions for indefinite singular equations by using Topological degree theory and some mathematical analysis skills, which may be of special interest. It is noted that our main methods can be studied other types of indefinite singular equations.
The following sections are organized as follows: In Section 2, we give some useful lemmas and definitions. In Section 3, main results are obtained for the existence of periodic wave solutions to the non-Newtonian filtration equation (1.1). In Section 4, two examples are given to show the feasibility of our results. Finally, some conclusions and discussions are given about this paper.

Preliminaries
Definition 2.1. [14] Let X and Z be two Banach spaces with norms || · || X , || · || Z , respectively. A continuous operator Let Ω ⊂ X be an open and bounded set with the origin θ ∈ Ω. N λ :Ω → Z, λ ∈ [0, 1] is said to be M-compact inΩ if there exists subset Z 1 of Z satisfying dimZ 1 = dimKerM and an operator R :Ω × [0, 1] → X 2 being continuous and compact such that for λ ∈ [0, 1], Let X and Z be two Banach spaces with norms || · || X , || · || Z , respectively. Let Ω ⊂ X be an open and bounded nonempty set. Suppose In addition, if the following conditions hold: Then the abstract equation Mx = N x has at least one solution in domM ∩Ω.
From Lemma 2.1, [28] and [29], we have the following lemma: Lemma 2.2. Consider the following p−Laplacian equation such that the following conditions hold.
(2) The equation Then Eq (2.1) has at least one T −periodic solution in Ω. Remark 2.1. Lemma 2.2 is derived from the Lemma 2.1 which is convenient for studying the existence of periodic wave solutions to the non-Newtonian filtration equation (1.1).

Main results
Denote Clearly, C T and C 1 T are Banach spaces. For each φ ∈ C T , let Consider the following equations family: Let Assume that the function f such that where f L and f M are positive constants. Furthermore, assume h > 0. Then for each u ∈ Ω, there are constants ξ 1 , ξ 2 ∈ [0, T ] such that Proof. Let u ∈ Ω, we have (3.1) holds. Dividing both sides of (3.1) by f (u) and integrating them on where we use f (u) > 0. By (3.2) and (3.3) we have By mean value theorem of integrals, there exists a point ξ 1 ∈ [0, T ] such that  Proof. We complete the proof by three steps.
Step 2. We will show that condition (2) of Lemma 2.2 is satisfied. On the contrary, assume that there exists u = a ∈ ∂Ω 1 such that F (a) = 0, then a ∈ R is a constant and We have which contradicts to a ∈ ∂Ω 1 . Hence, condition (2) of Lemma 2.2 is satisfied.
Step 3. We will show that condition (3) of Lemma 2.2 is satisfied. Due to the proof of Step 2, if u ∈ Ω 1 ∩ R such that F (u) = 0, the u = a ∈ [B 2 , B 1 ]. It is easy to see that a is unique by using f (u) is strictly monotonically increasing for u ∈ [B 2 , B 1 ]. Hence, Applying Lemma 2.2, we reach the conclusion. Lemma 3.2. Assume that the function f such that   By (3.14) and mean value theorem of integrals, there exists a point η 2 ∈ [0, T ] such that Thus, where h L = min t∈[0,T ] |h(t)|. For u ∈ Ω, by Lemma 3.2 and Hölder inequality we have where q > 1 and 1 p + 1 q = 1. Multiply (3.1) with u(t), and integrate it over the interval [0, T ], then In view of (3.16) and (3.19), we gain (3.20) In view of (3.1), (3.20) and (3.15), we have The following proof is similar to the proof of Step 2 and Step 3 in Theorem 3.1, we omit it. Remark 3.1. In Theorems 3.1 and 3.2, nonlinear term f (u) has no singularity at u = 0. For example, in Then, nonlinear term f (u) has singularity at u = 0. We naturally ask the following question: if nonlinear term f (u) has singularity at u = 0. i.e., lim u→0 + f (u) = ±∞, are there periodic wave solutions for Eq (1.1)? We very hope that the researchers will be able to solve the above problems. Remark 3.2. In [10], the authors studied the existence of periodic wave solutions for Eq (1.2) which nonlinear term f (q) is a strictly monotone function. Since monotonicity of f (q) is very critical for prior bounds of solutions, in the present paper, we also assume that f (q) is a strictly monotone function. When f (q) is not a monotone function, whether Eq (1.1) has periodic wave solutions which is a open problem. The above issue is our research topic. |e(s)| < +∞, then Eq (1.2) has at least one 2T −periodic wave solution. In the present paper, since Eq (1.1) has an indefinite singularity, we add the stronger conditions for nonlinear term f , i.e., assume that the function f such that where f L and f M are positive constants.
In this section, we will give two examples to illustrate the theoretical results in the present paper. Example 4.1. Consider the following non-Newtonian filtration equations with an indefinite singularity: Let y(t, x) = u(s) with s = x + ct be the solution of Eq (4.1), then Eq (4.1) is changed into the following equation: Obviously, f (u) = 1 1+u 2 > 0 is a strictly monotone increasing function. After a simple calculation, we have Thus, all conditions of Theorems 3.1 hold. Therefore, Theorems 3.1 guarantees the existence of at least one one periodic solution for Eq (4.2), i.e., Eq (4.1) has least one one periodic wave solution.
Then f (u) is a strictly monotone decreasing function. Thus, all conditions of Theorems 3.2 hold. and Theorems 3.2 guarantees the existence of at least one one periodic solution for Eq (4.2), i.e., Eq (4.1) has least one one periodic wave solution.

Conclusion
In this article, we study a non-Newtonian filtration equations with an indefinite singularity. By using an generalization of Mawhin's continuation theorem and some mathematic analysis methods, we obtain some existence results of periodic wave solutions for the considered equation. Two examples are used to demonstrate the usefulness of our theoretical results. The novelty of the present paper is that it is the first time to discuss the existence of periodic wave solutions for the indefinite singular non-Newtonian filtration equations. Our results improve and extend some corresponding results in the literature. However, many important questions about indefinite singular non-Newtonian filtration equations remain to be studied, such as oscillation problems, exponential stability and asymptotic stability problems, non-Newtonian filtration equations with impulse effects and stochastic effects, etc. We hope to focus on the above issues in future studies.