Citation: Famei Zheng. Periodic wave solutions of a non-Newtonian filtration equation with an indefinite singularity[J]. AIMS Mathematics, 2021, 6(2): 1209-1222. doi: 10.3934/math.2021074
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In this paper, we consider the periodic wave solutions problem for a type of non-Newtonian filtration equation with an indefinite singularity as follows:
∂y∂t=∂∂x(|∂y∂x|p−2∂y∂x)+f(y)+h(t,x)ym, | (1.1) |
where p>1,m>0,f∈C(R,R),h∈C(R×R,R). In this equation, the function 1ym may have a singularity at y=0. Besides this, the signs of h(t,x) are all allowed to change.
Equation (1.1) is known as the evolutionary p−Laplacian. Many fluid dynamics models can be described by Eq (1.1), see [1,2]. For the last forty years, there exist many results about non-Newtonian filtration equation. In 1967, Ladyzhenskaja [1] studied the following non-Newtonian filtration equation:
∂y∂t=∂∂x(|∂y∂x|p−2∂y∂x)+yq(1−y)(z−a),t≥0,p>1,x∈R |
which is the description of incompressible fluids and solvability in the large boundary value. Jin and Yin [3] investigated the traveling wavefronts for a non-Newtonian filtration equation with Hodgkin-Huxley source:
∂y∂t=∂∂x(|∂y∂x|p−2∂y∂x)+f(y,yτ),t≥0,x∈R, |
where p>1,f(y,z)=yq(1−y)(z−a),q>0,a∈(0,1) is a constant, yτ=y(x,t−τ),τ>0. The more related papers for non-Newtonian filtration equation, see e.g., [4,5,6,7].
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] studied the following non-Newtonian filtration equation:
∂q∂t=∂∂x(|∂q∂x|p−2∂q∂x)+f(q)+g(t,x). | (1.2) |
By using an extension of Mawhin's continuation theorem, the authors obtained some existence results of solitary wave and periodic wave solutions for Eq (1.2). Kong etc. [9] considered a non-Newtonian filtration equations with nonlinear sources and the variable delay. In 2017, when f(q) has a singularity (including the attractive singular case or the repulsive singular case) in (1.2), Lian etc. [10] studied the existence and multiplicity of positive periodic wave solutions for Eq (1.2). For more results about periodic solutions and periodic wave solutions, see [11,12,13].
In Eq (1.1), the signs of function h are allowed to change which means that the singularity of 1ym 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 second-order 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:
cu′(s)=(ϕp(u′(s)))′+f(u)−h(s)um, | (1.3) |
where ϕp(u)=|u|p−2u,p>1,m>0,f,h∈C(R,R).
Definition 1.1 Let T>0 be a constant. Suppose that u(s+T)=u(s) and u(s) is a solution of Eq (1.3) for s∈R. In generally, the periodic solution of Eq (1.3) is regarded as periodic wave solution of Eq (1.1).
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.
Definition 2.1.[14] Let X and Z be two Banach spaces with norms ||⋅||X,||⋅||Z, respectively. A continuous operator
M:X∩domM→Z |
is called to be quasi-linear if
(i) ImM:=M(X∩domM) is a closed subset of Z;
(ii) KerM:={x∈X∩domM:Mx=0} is linearly homeomorphic to Rn,n<∞,
where domM is the domain of M.
Definition 2.2.[14] Let Ω⊂X be an open and bounded set with the origin θ∈Ω. Nλ:ˉΩ→Z,λ∈[0,1] is said to be M-compact in ˉΩ if there exists subset Z1 of Z satisfying dimZ1=dimKerM and an operator R:ˉΩ×[0,1]→X2 being continuous and compact such that for λ∈[0,1],
(a) (I−Q)Nλ(ˉΩ)⊂ImM⊂(I−Q)Z,
(b) QNλx=0,λ∈(0,1)⇔QNx=0, ∀x∈Ω,
(c) R(⋅,0)≡0 and R(⋅,λ)|Σλ=(I−P)|Σλ,
(d) M[Pu+R(⋅,λ)]=(I−Q)Nλ,λ∈[0,1],
where X2 is a the complement space of KerM in X, i.e., X=KerM⊕X2; P,Q are two projectors satisfying ImP=KerM,ImQ=Z1, N=N1,Σλ={x∈ˉΩ:Mx=Nλx}.
Lemma 2.1.[14] Let X and Z be two Banach spaces with norms ||⋅||X,||⋅||Z, respectively. Let Ω⊂X be an open and bounded nonempty set. Suppose
M:X∩domM→Z |
is quasi-linear and Nλ:ˉΩ→Z,λ∈[0,1] is M−compact in ˉΩ. In addition, if the following conditions hold:
(A1) Mx≠Nλx,∀(x,λ)∈∂Ω×(0,1);
(A2) QNx≠0,∀x∈KerM∩∂Ω;
(A3) deg{JQN,Ω∩KerM,0}≠0,J:ImQ→KerMisahomeomorphism.
Then the abstract equation Mx=Nx has at least one solution in domM∩ˉΩ.
From Lemma 2.1, [28] and [29], we have the following lemma:
Lemma 2.2. Consider the following p−Laplacian equation
(ϕp(u′(s)))′=f(t,u,u′), | (2.1) |
where p>1,f∈C(R3,R) with f(t+T,⋅,⋅)=f(t,⋅,⋅). Assume that Ω is an open bounded set in C1T such that the following conditions hold.
(1) For each λ∈(0,1), the problem
(ϕp(u′))′=λf(t,u,u′),u(0)=u(T),u′(0)=u′(T), |
has no solution on ∂Ω.
(2) The equation
F(a)=1T∫T0f(t,a,0)dt=0, |
has no solution on ∂Ω∩R.
(3) The Brouwer degree
dB(F,Ω∩R,0)≠0. |
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).
Denote
CT={x|x∈C(R,R),x(t+T)≡x(t),∀t∈R} |
with the norm
|φ|0=maxt∈[0,T]|φ(t)|,∀φ∈CT |
and
C1T={x|x∈C1(R,R),x(t+T)≡x(t),∀t∈R} |
with the norm
|φ|∞=maxt∈[0,T]{|φ|0,|φ′|0},∀φ∈C1T. |
Clearly, CT and C1T are Banach spaces. For each ϕ∈CT, let
ϕ+(t)=max{ϕ(t),0},ϕ−(t)=max{−ϕ(t),0}, |
¯ϕ=1T∫T0ϕ(s)ds,||ϕ|p=(∫T0|ϕ(s)|pds)1p,p>1. |
Clearly, for t∈R, ϕ(t)=ϕ+(t)−ϕ−(t),¯ϕ=¯ϕ+−¯ϕ−. Consider the following equations family:
(ϕp(u′(s)))′=cλu′(s)−λf(u)+λh(s)um,λ∈(0,1]. | (3.1) |
Let
Ω={u∈C1T:(ϕp(u′(s)))′=cλu′(s)−λf(u)+λh(s)um,λ∈(0,1],u>0}. |
Lemma 3.1. Assume that the function f such that
fL<f(u)<fM,f′(u)>0,∀u>0, |
where fL and fM are positive constants. Furthermore, assume ¯h>0. Then for each u∈Ω, there are constants ξ1,ξ2∈[0,T] such that
u(ξ1)≤(¯h+fL)1m:=A1 |
and
u(ξ2)≥(¯hfM)1m:=A2. |
Proof. Let u∈Ω, we have (3.1) holds. Dividing both sides of (3.1) by f(u) and integrating them on [0,T], we have
∫T0(ϕp(u′(s)))′f(u)ds=−λT+λ∫T0h(s)f(u)umds,λ∈(0,1]. | (3.2) |
Note that
∫T0(ϕp(u′(s)))′f(u)ds=∫T01f(u)dϕp(u′(s))=∫T0f−2(u)f′(u)|u′|pds≥0, | (3.3) |
where we use f′(u)>0. By (3.2) and (3.3) we have
T≤∫T0h(s)f(u)umds≤∫T0h+(s)fLumds. |
By mean value theorem of integrals, there exists a point ξ1∈[0,T] such that
um(ξ1)≤¯h+fL, |
i.e.,
u(ξ1)≤(¯h+fL)1m:=A1. |
Multiplying both sides of (3.1) by um and integrating them on [0,T], we have
∫T0(ϕp(u′(s)))′umds=−λ∫T0f(u)umds+λ∫T0h(s)ds,λ∈(0,1]. | (3.4) |
Note that
∫T0(ϕp(u′(s)))′umds=∫T0umdϕp(u′(s))=−m∫T0um−1|u′|pds≤0. | (3.5) |
In view of (3.4) and (3.5), we have
∫T0f(u)umds≥∫T0h(s)ds=T¯h |
and
fM∫T0umds≥T¯h. |
By mean value theorem of integrals, there exists a point ξ2∈[0,T] such that
um(ξ2)≥¯hfM, |
i.e.,
u(ξ2)≥(¯hfM)1m:=A2. |
Theorem 3.1. Suppose that conditions of Lemma 3.1 hold. Further assume that some assumptions on f(u) and h(t):
(H1) Suppose that f(u)≤1um+1 for u>0 and m>0, and h(t)>0 for t∈R.
Then Eq (1.3) has at least one T−periodic solution, i.e., Eq (1.1) has at least one periodic wave solution, if c<0 and Am+12−m+1|c|TfMBm1−m+1|c|T¯|h|>0, where A2 is defined by Lemma 3.1, B1 is defined by (3.6).
Proof. We complete the proof by three steps.
Step 1. For t1<t2, let
u(t1)=maxt∈[0,T]u(t),u(t2)=mint∈[0,T]u(t). |
By Eq (3.1) and (ϕp(u′(t)))′|t=t1≤0, we have
f(u(t1))≥h(t1)um(t1). |
Thus, by assumption (H1) we have
u(t1)≤1hL:=B1, | (3.6) |
where hl=mint∈[0,T]|h(t)|. Multiplying both sides of (2.2) by um and integrating them on [t1,t2], we have
∫t2t1(ϕp(u′(s)))′umds=cλ∫t2t1u′(s)umds−λ∫t2t1f(u)umds+λ∫t2t1h(s)ds,λ∈(0,1]. | (3.7) |
Obviously, ∫t2t1(ϕp(u′(s)))′umds≤0. Then, from (3.7), Lemma 3.1 and assumptions of Theorem 3.1, we have
c∫t2t1u′(s)umds−∫t2t1f(u)umds+∫t2t1h(s)ds≤0 |
and
um+1(t2)≥um+1(t1)+m+1c∫t2t1f(u)umds−m+1c∫t2t1h(s)ds≥Am+12−m+1|c|TfMBm1−m+1|c|T¯|h|. |
Since Am+12−m+1|c|TfMBm1−m+1|c|T¯|h|>0, we get
u(t2)≥(Am+12−m+1|c|TfMBm1−m+1|c|T¯|h|)1m+1:=B2. | (3.8) |
Multiplying both sides of (3.1) by u′(t) and integrating them on [0,T], we have
∫T0(ϕp(u′(s)))′u′(s)ds=cλ∫T0|u′(s)|2ds−λ∫T0f(u)u′(s)ds+λ∫T0h(s)umu′(s)ds. | (3.9) |
Obviously,
∫T0(ϕp(u′(s)))′u′(s)ds=∫T0u′(s)dϕp(u′(s))=0 | (3.10) |
and
∫T0f(u)u′(s)ds=0. | (3.11) |
From (3.9)–(3.11), we get
||u′||22≤1|c|Bm1∫T0|h(s)||u′(s)|ds≤1|c|Bm1||h||2||u′||2, |
i.e.,
||u′||2≤1|c|Bm1||h||2. | (3.12) |
In view of (3.1), (3.12) and H¨older inequality, we have
|u′(t)|p−1=|ϕp(u′(t1)+∫tt1(ϕp(u′(s))′ds|≤∫T0|(ϕp(u′(s))′|ds≤∫T0|c||u′(s)|ds+∫T0|f(u(s))|ds+∫T0|h(s)|um(s)ds≤|c|T12||u′(s)||2+TfM+T¯|h|Bm2≤T12Bm1||h||2+TfM+T¯|h|Bm2:=M1, |
i.e.,
|u′|0≤M1p−11. |
Choose positive constants δ1,δ2 and M such that δ1<B2<B1<δ2 and M>M1p−11. Let
Ω1={u∈C1T:δ1<u(t)<δ2,|u′(t)|<M}. |
For each λ∈(0,1), Eq (3.1) has no solution on ∂Ω1. Hence, condition (1) of Lemma 2.2 is satisfied.
Step 2. We will show that condition (2) of Lemma 2.2 is satisfied. On the contrary, assume that there exists u=a∈∂Ω1 such that F(a)=0, then a∈R is a constant and
F(a)=1T∫T0[−f(a)+h(s)am]=0. |
We have
B2≤(¯hfM)1m≤a≤(hMfL)1m≤B1 |
which contradicts to a∈∂Ω1. 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∈Ω1∩R such that F(u)=0, the u=a∈[B2,B1]. It is easy to see that a is unique by using f(u) is strictly monotonically increasing for u∈[B2,B1]. Hence,
dB(F,Ω∩R,0)=1≠0. |
Applying Lemma 2.2, we reach the conclusion.
Lemma 3.2. Assume that the function f such that
f(0)=limu→0+f(u)>0,f(u)>0,f′(u)<0,∀u>0. |
Furthermore, assume ¯h>0. Then for each u∈Ω, there are constants η1,η2∈[0,T] such that
u(η1)≤(¯h+¯f)1m:=A3 |
and
u(η2)≥(¯hf(0))1m:=A4. |
Proof. Integrating (3.1) on [0,T], we have
∫T0f(u)ds=∫T0h(s)umds |
and
T¯f≤∫T0h+(s)umds. | (3.13) |
By (3.13) and mean value theorem of integrals, there exists a point η1∈[0,T] such that
um(η1)≤¯h+¯f, |
i.e.,
u(η1)≤(¯h+¯f)1m:=A3. |
By f′(u)<0 for u>0, we have f(0)>f(u) for u>0. Similar to the proof of (3.4) and (3.5) in Lemma 3.1, we have
∫T0f(u)umds≥T¯h |
and
f(0)∫T0umds≥T¯h. | (3.14) |
By (3.14) and mean value theorem of integrals, there exists a point η2∈[0,T] such that
um(η2)≥¯hf(0), |
i.e.,
u(η2)≥(¯hf(0))1m:=A4. |
Theorem 3.2. Suppose that conditions of Lemma 3.2 hold. Then Eq (1.3) has at least one T−periodic solution, i.e., Eq (1.1) has at least one periodic wave solution.
Proof. Let u(t0)=mint∈[0,T]u(t). By Eq (3.1), we have
f(u(t0))=h(t0)um(t0). |
Thus,
u(t0)≥(hLf(0))1m:=B0, | (3.15) |
where hL=mint∈[0,T]|h(t)|. For u∈Ω, by Lemma 3.2 and H¨older inequality we have
|u|0≤A3+T1q(∫T0|u′(s)|pds)1p, | (3.16) |
where q>1 and 1p+1q=1. Multiply (3.1) with u(t), and integrate it over the interval [0,T], then
∫T0|u′(s)|pds=λ∫T0f(u)uds−λ∫T0h(s)umuds≤∫T0f(u)uds+∫T0h−(s)umuds≤|u|0∫T0f(u)ds+|u|0∫T0h−(s)umds. | (3.17) |
Integrating (3.1) over the interval [0,T], we gain
∫T0f(u)ds=∫T0h(s)umds. | (3.18) |
By (3.17) and (3.18), we have
∫T0|u′(s)|pds≤|u|0∫T0h+(s)umds≤T|u|0¯h+Bm0. | (3.19) |
In view of (3.16) and (3.19), we gain
|u|0≤A3+T1q(T|u|0¯h+Bm0)1p |
which implies that there is a constant ρ>0 such that
|u|0<ρ, |
i.e.,
maxt∈[0,T]u(t)<ρ. |
From (3.9)–(3.11) and (3.15), we have
||u′||2≤1|c|Bm0||h||2. | (3.20) |
In view of (3.1), (3.20) and (3.15), we have
|u′(t)|p−1=|ϕp(u′(t0)+∫tt0(ϕp(u′(s))′ds|≤∫T0|(ϕp(u′(s))′|ds≤∫T0|c||u′(s)|ds+∫T0f(u(s))ds+∫T0|h(s)|um(s)ds≤|c|T12||u′(s)||2+T¯f+T¯|h|Bm0≤T12Bm0||h||2+T¯f+T¯|h|Bm0:=N, |
i.e.,
|u′|0≤N1p−1. |
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 Eq (1.1), let f(u)=1u2 or f(u)=−1u2. 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., limu→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.
Remark 3.3. In [8], Eq (1.2) is changed into the following equation:
−cu′(s)=(ϕp(u′(s)))′+f(u(s))+e(s). |
Under the following assumptions:
(H1) there exist constants m0>0,m1>1 such that
uf(u)≤−m0um,∀u∈R, |
(H2) e∈C(R,R) is a continuous 2T−periodic function with e(s)≠0, and
(∫t−T|e(s)|mm−1)m−1m+sups∈[−T,T]|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
fL<f(u)<fM,f′(u)>0,∀u>0, |
where fL and fM 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:
∂y∂t=∂∂x(|∂y∂x|p−2∂y∂x)+f(y)+h(t,x)ym. | (3.21) |
Let 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 (4.1), then Eq (4.1) is changed into the following equation:
cu′(s)=(ϕp(u′(s)))′+f(u)−h(s)um. | (3.22) |
Let p=3,m=1,T=2π,f(u)=1+arctanu,h(s)=1+sins,c=−260. Obviously, f′(u)=11+u2>0 is a strictly monotone increasing function. After a simple calculation, we have
fL=1,fM=1+π2,¯h=1,hM=2,A2=(¯hfM)1m≐0.39, |
B1=(hMfL)1m=1,Am+12−m+1|c|TBm1−m+1|c|T¯|h|=0.05538>0. |
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.
Example 4.2. In Eq (4.2), let p=3,m=1,T=2π,f(u)=3−arctanu,h(s)=1+sins. Obviously,
¯h=1>0,f(0)=3>0,f(u)>0,f′(u)=−11+u2<0foru>0. |
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.
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.
The authors would like to express the sincere appreciation to the editor and reviewers for their helpful comments in improving the presentation and quality of the paper.
The author confirms that they have no conflict of interest.
[1] | O. Ladyzhenskaja, New equation for the description of incompressible fluids and solvability in the large boundary value of them, P. Steklov I. Math., 102 (19677), 95-118. |
[2] | L. Martinson, K. Pavlov, Magnetohydrodynamics of non-Newtonian fluids, Magnetohydrodynamics, 11 (1975), 47-53. |
[3] |
C. Jin, J. Yin, Traveling wavefronts for a time delayed non-Newtonian filtration equation, Physica D, 241 (2012), 1789-1803. doi: 10.1016/j.physd.2012.08.007
![]() |
[4] |
Z. Fang, X. Xu, Extinction behavior of solutions for the p-Laplacian equations with nonlocal source, Nonlinear Anal. Real, 13 (2012), 1780-1789. doi: 10.1016/j.nonrwa.2011.12.008
![]() |
[5] | T. Zhou, B. Du, H. Du, Positive periodic solution for indefinite singular Lienard equation with p-Laplacian, Adv. Differ. Equ., 158 (2019), 1-12. |
[6] | S. Ji, J. Yin, R. Huang, Evolutionary p-Laplacian with convection and reaction under dynamic boundary condition, Bound. Value Probl., 194 (2015), 1-12. |
[7] |
F. Sanchez-Garduno, P. Maini, Existence and uniqueness of a sharp front travelling wave in degenerate nonlinear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178
![]() |
[8] |
Z. Liang, J. Chu, S. Lu, Solitary wave and periodic wave solutions for a non-Newtonian filtration equation, Math. Phys. Anal. Geom., 17 (2014), 213-222. doi: 10.1007/s11040-014-9150-9
![]() |
[9] |
F. Kong, Z. Luo, Solitary wave and periodic wave solutions for the non-Newtonian filtration equations with non- linear sources and a time-varying delay, Acta Math. Sci., 37 (2017), 1803-1816. doi: 10.1016/S0252-9602(17)30108-X
![]() |
[10] |
Z. Liang, F. Kong, Positive periodic wave solutions of singular non-Newtonian filtration equations, Anal. Math. Phys., 7 (2017), 509-524. doi: 10.1007/s13324-016-0153-5
![]() |
[11] | H. Yin, B. Du, Stochastic patch structure Nicholson's blowfies system with mixed delays, Adv. Differ. Equ., 386 (2020), 1-11. |
[12] | H. Yin, B. Du, Q. Yang, F. Duan, Existence of homoclinic orbits for a singular differential equation involving p-Laplacian, J. Funct. Space., 2020 (2020), 1-7. |
[13] |
S. Lu, Periodic solutions to a second order p-Laplacian neutral functional differential system, Nonlinear Anal., 69 (2008), 4215-4229. doi: 10.1016/j.na.2007.10.049
![]() |
[14] |
W. Ge, J. Ren, An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-Laplacain, Nonlinear Anal., 58 (2004), 477-488. doi: 10.1016/j.na.2004.01.007
![]() |
[15] |
R. Hakl, M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differ. Equations, 263 (2017), 451-469. doi: 10.1016/j.jde.2017.02.044
![]() |
[16] | A. Fonda, A. Sfeccib, On a singular periodic Ambrosetti-Prodi problem, Nonlinear Anal., 19 (2017), 146-155. |
[17] |
S. Kumar, D. Kumar, Solitary wave solutions of (3+1)-dimensional extended Zakharov-Kuznetsov equation by Lie symmetry approach, Comput. Math. Appl., 77 (2019), 2096-2113. doi: 10.1016/j.camwa.2018.12.009
![]() |
[18] |
D. Kumar, S. Kumar, Some new periodic solitary wave solutions of (3+1)-dimensional generalized shallow water wave equation by Lie symmetry approach, Comput. Math. Appl., 78 (2019), 857-877. doi: 10.1016/j.camwa.2019.03.007
![]() |
[19] |
S. Kumar, D. Kumar, Lie symmetry analysis and dynamical structures of soliton solutions for the (2+1)-dimensional modified CBS equation, Int. J. Mod. Phys. B, 34 (2020), 2050221. doi: 10.1142/S0217979220502215
![]() |
[20] |
S. Kumar, D. Kumar, Lie symmetry reductions and group Invariant Solutions of (2+1)-dimensional modified Veronese web equation, Nonlinear Dynam., 98 (2019), 1891-1903. doi: 10.1007/s11071-019-05294-x
![]() |
[21] |
D. Kumar, S. Kumar, Solitary wave solutions of pZK equation using Lie point symmetries, Eur. Phys. J. Plus, 135 (2020), 162. doi: 10.1140/epjp/s13360-020-00218-w
![]() |
[22] | S. Kumar, M. Niwas, Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2+1)-dimensional NNV equations, Phys. Scripta, 95 (2020), 095204. |
[23] | S. Rani, Lie symmetry reductions and dynamics of soliton solutions of (2+1)-dimensional Pavlov equation, Pramana, 19 (2020), 116. |
[24] |
S. Kumar, A. Kumar, H. Kharbanda, Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations, Phys. Scripta, 95 (2020), 065207. doi: 10.1088/1402-4896/ab7f48
![]() |
[25] | S. Lu, X. Yu, Periodic solutions for second order differential equations with indefinite singularities, Adv. Nonlinear Anal., 9 (2020), 994-1007. |
[26] | S. Lu, R. Xue, Periodic solutions for a Liénard equation with indefinite weights, Topol. Method. Nonlinear Anal., 54 (2019), 203-218. |
[27] |
S. Lu, Y. Guo, L. Chen, Periodic solutions for Liénard equation with an indefinite singularity, Nonlinear Anal. Real, 45 (2019), 542-556. doi: 10.1016/j.nonrwa.2018.07.024
![]() |
[28] | B. Du, S. Lu, On the existence of periodic solutions to a p-Laplacian equation, Indian J. Pure Appl. Math., 40 (2009), 253-266. |
[29] | Y. Xin, Z. Chen, Positive periodic solution for prescribed mean curvature generalized Lienard equation with a singularity, Bound. Value Probl., 89 (2020), 1-15. |
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