Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 05, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu TRAVELING WAVES WITH SINGULARITIES IN A DAMPED HYPERBOLIC MEMS TYPE EQUATION IN THE PRESENCE OF NEGATIVE POWERS NONLINEARITY YU ICHIDA Abstract. We consider traveling waves with singularities in a damped hyper- bolic MEMS type equation in the presence of negative powers nonlinearity. We investigate how the existence of traveling waves, their shapes, and asymptotic behavior change with the presence or absence of an inertial term. These are studied by applying the framework that combines Poincaré compactification, classical dynamical systems theory, and geometric methods for the desingular- ization of vector fields. We report that the presence of this term causes the shapes to change significantly for sufficiently large wave speeds. 1. Introduction In this paper, we consider the following damped hyperbolic MEMS type equation with negative powers nonlinearity, ε2utt + ut = uxx + (1− u)−α, t > 0, x ∈ R, (1.1) where α ∈ 2N and ε > 0. Here, ε is a small constant and the ratio of the interaction due to the inertial and damped terms (see [5, 7, 8, 9] and references therein). Equation (1.1) is based on the equation ut = uxx + (1− u)−α, t > 0, x ∈ R, α ∈ N (1.2) treated in [10, 15], with the term ε2utt added to the left-hand side. (1.1) is a type of partial differential equation commonly referred to as a damped hyperbolic equation. Since (1.1) has aspects of both parabolic and hyperbolic types, it has re- cently attracted attention from the viewpoint of partial differential equation theory (see [7, 6]). Guo [7] considers both parabolic and hyperbolic type problem about MEMS, and provides some quenching criteria. For MEMS, see below. In addition, it discusses the global existence of solutions. The previous work [6] is concerned with the behavior of the solutions to the nonlinear damped hyperbolic Allen-Cahn equation with appropriate boundary conditions and initial data in a bounded do- main. They argue that reaction-diffusion equations have the lack of inertial and others. There are many ways to overcome these unphysical properties; one of them is to consider hyperbolic reaction-diffusion equations. 2020 Mathematics Subject Classification. 34C05, 34C08, 35B40, 35C07, 35L81, 74H35. Key words and phrases. MEMS type equation; Poincaré compactification; Desingularization of vector fields (blow-up); Asymptotic behavior. ©2023. This work is licensed under a CC BY 4.0 license. Submitted June 22, 2022. Published January 16, 2023. 1 2 Y. ICHIDA EJDE-2023/05 Furthermore, (1.1) and (1.2) are special cases of the generalized MEMS type partial differential equation (see [5, 8, 12] and references therein). The MEMS model is the Micro-Electro Mechanical System devices and used in many machines around us (for instance, see [16]). In general, the MEMS model is known to induce the touchdown phenomenon (mathematically, quenching). Clarification of the structure of singularity formation, such as quench, is one of the most important issues in MEMS type equations, and there have been a lot of studies recently. However, since the nonlinear terms of MEMS equations are not simple, then they have both hyperbolic and parabolic aspects, it is not fully understood what kind of typical solutions exist. In this article, we investigate how the behavior (shapes and asymptotic behavior) of traveling waves change depending on whether the ε2utt term is present or absent in the left-hand side of [10]. More precisely, in the traveling wave framework, we compare the family of functions satisfying (1.1) with the family of functions satisfying (1.2) revealed in [10] in terms of the asymptotic behavior and shapes. The reason why we refer to the traveling waves as families of functions satisfying the equations is that they cause singularities at the endpoints of finite intervals despite the equations being defined over the whole domain, which makes subsequent analysis difficult (see [10]). In addition, we are interested in whether the asymptotic behavior obtained from (1.1) and that from (1.2) coincide as ε → 0. Although it appears to be nothing more than adding ε2utt to the left-hand side of (1.2), this extension allows us to obtain conclusions from the perspective of traveling waves that cannot be obtained in [10]. To the best of the author’s knowledge, there has been no analysis of the existence, shapes and asymptotic behavior of traveling waves in such a type of equation with both hyperbolic and parabolic forms. We believe that this paper will provide this abundant information through a dynamical systems approach and give a new perspective on these types of equations. To consider the traveling waves of (1.1), we introduce the change of variables φ(ξ) = 1− u(t, x), ξ = x− ct, 0 < c ∈ R. Then solving (1.1) reduces to solving for φ(ξ) in (1− ε2c2)φ′′ = −cφ′ + φ−α, (′= d dξ ). (1.3) Equation (1.3) with ε = 0 is discussed in [10]. In (1.3), there is a case classification for 1− ε2c2 that did not appear in [10]. When 1− ε2c2 = 0, i.e., c = 1/ε, from (1.3) we obtain the differential equation 0 = −cφ′ + φ−α. This can be solved by φ(ξ) = (α+ 1 c ξ +B ) 1 α+1 (1.4) with a constant B ∈ R. In other words, we can express φ(ξ) explicitly in this case. For a discussion of this case, see Remark 2.11. Hereinafter 1− ε2c2 6= 0. Then, (1.3) is equivalent to φ′ = ψ, ψ′ = (1− ε2c2)−1(−cψ + φ−α). (1.5) EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 3 In (1.5), the dynamics to infinity in the equation with ε = 0 has been studied in [10, 11]. In [11], although the partial differential equations are different, the ordinary differential equations (ODEs for short) derived from them include the ODEs of [10]. As can be seen from these previous studies, (1.5) is not easy to analyze. However, as shown in [10, 11, 14, 15], it is possible to study the dynamics of this ODE to infinity in the framework that combines Poincaré compactification (for instance, see [10, Section 2] and [4, 14, 15] for the details of it), classical dynamical systems theory, and geometric methods for desingularization of vector fields (see [4, Section 3] and references therein). By using these methods, the whole dynamics on the phase space R2 including infinity (denoted by Poincaré disk) generated by the two-dimensional differential equation (1.5) is obtained. In other words, from these dynamics, we expect to categorize all traveling waves as in these previous studies. Furthermore, the strength of the analysis in this framework is that the existence of connecting orbits in dynamical systems including infinity not only proves the existence of these traveling waves, provides information about their shapes but allows us to study their asymptotic behavior. This article is organized as follows. In the next section, we reproduce the termi- nology defined in [10] and the main results obtained, and state the main results of this paper. In Section 3, we obtain the dynamics of (1.5) on the Poincaré disk via Poincaré compactification and basic theory of the dynamical systems. The proof of Theorems will be completed in Section 4. Section 5 is devoted to the concluding remarks. 2. Known and main results Before we state the main results of this paper, we reproduce the following def- initions of quasi traveling waves and quasi traveling waves with quenching. The reason for this is that the main result in this paper will be compared later with that in [10] (see Proposition 2.3 and Theorem 2.5). Here, quenching in ODE (1.5) roughly means that the following holds φ(ξ)→ 0, |φ′(ξ)| → +∞, as ξ → |ξ∗| with |ξ∗| < +∞. Definition 2.1 (Definition 1, [10]). We say that a function u(t, x) ≡ 1− φ(ξ) is a quasi traveling wave of (1.2) if the function φ(ξ) is a solution of (1.3) with ε = 0 on a finite interval or semi-infinite interval. Definition 2.2 (Definition 2, [10]). We say that a function u(t, x) ≡ 1 − φ(ξ) is a quasi traveling wave with quenching of (1.2) if the function u(t, x) is a quasi traveling wave of (1.2) on a finite interval (resp. semi-infinite interval) such that φ reaches 0 and |φ′| becomes infinite at both ends of the interval (resp. finite end point of the semi-infinite interval). More precisely, we have the following three cases: (I) The function φ(ξ) is a solution of (1.3) with ε = 0 on a semi-infinite interval (−∞, ξ∗) (φ(ξ) ∈ C2(−∞, ξ∗) ∩ C0(−∞, ξ∗], |ξ∗| <∞), and satisfies lim ξ→ξ∗−0 φ(ξ) = 0 and lim ξ→ξ∗−0 |ψ(ξ)| =∞. (II) The function φ(ξ) is a solution of (1.3) with ε = 0 on a semi-infinite interval (ξ∗,+∞) (φ(ξ) ∈ C2(ξ∗,+∞) ∩ C0[ξ∗,+∞), |ξ∗| <∞), and satisfies lim ξ→ξ∗+0 φ(ξ) = 0 and lim ξ→ξ∗+0 |ψ(ξ)| =∞. 4 Y. ICHIDA EJDE-2023/05 (III) The function φ(ξ) is a solution of (1.3) with ε = 0 on a finite interval (ξ−, ξ+) (φ(ξ) ∈ C2(ξ−, ξ+) ∩ C0[ξ−, ξ+], −∞ < ξ− < ξ+ < +∞), and satisfies the following lim ξ→ξ+−0 φ(ξ) = 0, lim ξ→ξ−+0 φ(ξ) = 0, lim ξ→ξ+−0 |ψ(ξ)| =∞, lim ξ→ξ−+0 |ψ(ξ)| =∞. With these definitions, we review the results obtained in [10]. Note that the meaning of the symbol F (η) ∼ G(η) as η → +∞ is lim η→+∞ ∣∣F (η) G(η) ∣∣ = 1. Proposition 2.3 (Theorem 2, [10]). Assume that α ∈ 2N. Then (1.2) possesses a family of quasi traveling waves with quenching on a finite interval. Moreover, each quasi traveling wave with quenching u(t, x) = 1− φ(ξ) satisfies the following: • limξ→ξ+−0 φ(ξ) = 0, limξ→ξ−+0 φ(ξ) = 0, limξ→ξ+−0 ψ(ξ) =∞, limξ→ξ−+0 ψ(ξ) = −∞. • φ(ξ) < 0 holds for ξ ∈ (ξ−, ξ+). • There exists a constant ξ∗ ∈ (ξ−, ξ+) such that ψ(ξ) < 0 for ξ ∈ (ξ−, ξ∗), ψ(ξ∗) = 0, and ψ(ξ) > 0 for ξ ∈ (ξ∗, ξ+). In addition, the quenching rates are φ(ξ) ∼ −C(ξ+ − ξ) 2 α+1 ψ(ξ) ∼ C (ξ+ − ξ)− α−1 α+1 (2.1) as ξ → ξ+ − 0, and φ(ξ) ∼ −C(ξ − ξ−) 2 α+1 ψ(ξ) ∼ −C (ξ − ξ−) −α−1 α+1 (2.2) as ξ → ξ− + 0, with C > 0. Remark 2.4. Note that the asymptotic behavior for (2.1) and (2.2) in Proposition 2.3 differs in the exponential part from the asymptotic behavior obtained in [10, Theorem 2] and [11, Proposition 1]. The reason for this is that, after the publication of [10, 11], we chose more appropriate principal terms in the computational pro- cess of deriving the asymptotic behavior, which resulted in higher accuracy. This improvement is described in detail in Subsection 4.1. Furthermore, this improve- ment has already been introduced into [12], and the asymptotic behavior, which was previously difficult to derive, has been obtained. However, the underlying idea is similar to the previous ones. Next, the main results of this paper are described. Figures 1, 2, and 3 show the schematic pictures of traveling waves obtained by each theorem. Theorem 2.5. Assume that α ∈ 2N, ε > 0, and 1 − ε2c2 > 0. Then, for a given positive constant 0 < c < 1/ε, there exists a family of the functions (which corresponds to a family of the orbits of (1.5)) defined on the finite intervals such that each function u(t, x) satisfies equation (1.1) on a finite interval (ξ−, ξ+) (−∞ < ξ− < ξ+ < +∞). Moreover, each function u(t, x) ≡ 1−φ(ξ) satisfies the following: • limξ→ξ+−0 φ(ξ) = 0, limξ→ξ−+0 φ(ξ) = 0, limξ→ξ+−0 ψ(ξ) =∞, limξ→ξ−+0 ψ(ξ) = −∞. EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 5 • φ(ξ) < 0 holds for ξ ∈ (ξ−, ξ+). • There exists a constant ξ∗ ∈ (ξ−, ξ+) such that the following holds: ψ(ξ) < 0 for ξ ∈ (ξ−, ξ∗), ψ(ξ∗) = 0 and ψ(ξ) > 0 for ξ ∈ (ξ∗, ξ+). In addition, the asymptotic behavior for ξ → ξ+ − 0 and ξ → ξ− + 0 are same as (2.1) and (2.2). On the other hand, assume that 1−ε2c2 < 0. Then, for a given positive constant c > 1/ε, there exists a family of the functions (which corresponds to a family of the orbits of (1.5)) defined on the finite intervals such that each function u(t, x) satisfies equation (1.1) on a finite interval (ξ−, ξ+) (−∞ < ξ− < ξ+ < +∞). Moreover, each function u(t, x) ≡ 1− φ(ξ) satisfies the following: • limξ→ξ+−0 φ(ξ) = 0, limξ→ξ−+0 φ(ξ) = 0, limξ→ξ+−0 ψ(ξ) = −∞, limξ→ξ−+0 ψ(ξ) = +∞. • φ(ξ) > 0 holds for ξ ∈ (ξ−, ξ+). • There exists a constant ξ∗ ∈ (ξ−, ξ+) such that ψ(ξ) > 0 for ξ ∈ (ξ−, ξ∗), ψ(ξ∗) = 0 and ψ(ξ) < 0 for ξ ∈ (ξ∗, ξ+). In addition, the asymptotic behaviors are φ(ξ) ∼ C(ξ+ − ξ) 2 α+1 ψ(ξ) ∼ −C (ξ+ − ξ)− α−1 α+1 (2.3) as ξ → ξ+ − 0, and φ(ξ) ∼ C(ξ − ξ−) 2 α+1 ψ(ξ) ∼ C (ξ − ξ−) −α−1 α+1 (2.4) as ξ → ξ− + 0, with C > 0. ξ u = 1− φ ξ+ξ− u = 1 0 ξ u = 1− φ ξ+ξ− u = 1 0 Figure 1. Schematic picture of the functions defined on the finite interval such that each function u(t, x) ≡ 1−φ(ξ) satisfies equation (1.1) on a finite interval (ξ−, ξ+) in Theorem 2.5. Here it should be noted that the position of the singular points ξ− and ξ+ are not determined in our studies, however, they are shown in this figure for the convenience. [Left: In the case that 1 − ε2c2 > 0.] [Right: In the case that 1 − ε2c2 < 0.] Note that in the figure on the right, the trajectory in which the minimum of u is below the ξ-axis is chosen from among the infinitely many trajectories that correspond to Theorem 2.5. 6 Y. ICHIDA EJDE-2023/05 Remark 2.6. In Theorem 2.5, the result for the case 1 − ε2c2 > 0 is almost the same as in Proposition 2.3. However, since (1.1) is a hyperbolic equation and there is room for consideration in adopting Definition 2.2 as a rigorous discussion of the mathematical formulation of the solution, Theorem 2.5 is phrased as the existence of a family of functions satisfying the equation. Notable points in Theorem 2.5 are as follows: (i) In addition, the asymptotic behavior obtained in the above theorem is the same as Proposition 2.3, except for the difference in the sign of the coefficients. This means that the behavior of these as ε → 0 does not change. This may be due to the fact that the principal part of the derived vector field (3.2) does not change. (ii) The most important point to be emphasized in this result is that a condition on the wave speed that is not obtained in [10] appears, and when the wave speed exceeds c = 1/ε, that is, when the wave speed is sufficiently large, traveling waves that are not seen in [10] are observed (see Figure 1 and [10, Figure 1]). Theorem 2.7. Assume that α ∈ 2N, ε > 0, and 1 − ε2c2 > 0. Then, for a given positive constant 0 < c < 1/ε, there exists a family of the functions (which cor- responds to a family of the orbits of (1.5)) defined on the semi-infinite intervals such that each function u(t, x) satisfies (1.1) on a semi-infinite interval (−∞, ξ+) (−∞ < ξ+ < +∞). Moreover, each function u(t, x) ≡ 1− φ(ξ) satisfies the follow- ing: • limξ→ξ+−0 φ(ξ) = 0, limξ→−∞ φ(ξ) = −∞, limξ→ξ+−0 ψ(ξ) =∞. • φ(ξ) < 0 holds for ξ ∈ (−∞, ξ+). In addition, the asymptotic behavior for ξ → ξ+ − 0 and ξ → −∞ are (2.1) and φ(ξ) ∼ −Ce− c 1−ε2c2 ξ asξ → −∞ (2.5) with C > 0. On the other hand, assume that 1−ε2c2 < 0. Then, for a given positive constant c > 1/ε, there exists a family of the functions (which corresponds to a family of the orbits of (1.5)) defined on the semi-infinite intervals such that each function u(t, x) satisfies (1.1) on a semi-infinite interval (ξ−,+∞) (−∞ < ξ− < +∞). Moreover, each function u(t, x) ≡ 1− φ(ξ) satisfies the following: • limξ→ξ−+0 φ(ξ) = 0, limξ→∞ φ(ξ) = +∞, limξ→ξ−+0 ψ(ξ) = +∞. • φ(ξ) > 0 holds for ξ ∈ (ξ−,+∞). In addition, the asymptotic behavior for ξ → ξ− + 0 and ξ → +∞ are (2.4) and φ(ξ) ∼ Ce− c 1−ε2c2 ξ as ξ → +∞ (2.6) with C > 0. Theorem 2.8. Assume that α ∈ 2N, ε > 0, and 1 − ε2c2 > 0. Then, for a given positive constant 0 < c < 1/ε, there exists a family of functions (which corresponds to a family of the orbits of (1.5)) defined on the semi-infinite intervals such that each function u(t, x) satisfies (1.1) on a semi-infinite interval (−∞, ξ+) (−∞ < ξ+ < +∞). Moreover, each function u(t, x) ≡ 1− φ(ξ) satisfies the following: • limξ→ξ+−0 φ(ξ) = 0, limξ→−∞ φ(ξ) = −∞, limξ→ξ+−0 ψ(ξ) =∞. • φ(ξ) < 0 holds for ξ ∈ (−∞, ξ+). EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 7 In addition, the asymptotic behavior for ξ → ξ+ − 0 and ξ → −∞ are (2.1) and φ(ξ) ∼ O(ξ 1 α+1 ), ψ(ξ) ∼ O((−ξ)− α α+1 ), (2.7) as ξ → −∞. On the other hand, assume that 1−ε2c2 < 0. Then, for a given positive constant c > 1/ε, there exists a family of the functions (which corresponds to a family of the orbits of (1.5)) defined on the semi-infinite intervals such that each function u(t, x) satisfies equation (1.1) on a semi-infinite interval (ξ−,+∞) (−∞ < ξ− < +∞). Moreover, each function u(t, x) ≡ 1− φ(ξ) satisfies the following: • limξ→ξ−+0 φ(ξ) = 0, limξ→+∞ φ(ξ) = +∞, limξ→ξ−+0 ψ(ξ) = +∞. • φ(ξ) > 0 holds for ξ ∈ (ξ−,+∞). In addition, the asymptotic behavior for ξ → ξ− + 0 and ξ → +∞ are (2.4) and φ(ξ) ∼ O(ξ 1 α+1 ), ψ(ξ) ∼ O((−ξ)− α α+1 ), (2.8) as ξ → +∞. ξ u = 1− φ ξ+ u = 1 0 ξ u = 1− φ u = 1 0 ξ− Figure 2. Schematic pictures of the functions defined on the semi- infinite interval such that each function u(t, x) ≡ 1− φ(ξ) satisfies equation (1.1) on a semi-infinite interval in Theorem 2.7 and The- orem 2.8. Here it should be noted that the position of the singular point ξ+ (or ξ−) are not determined in our studies, however, they are shown in these figures for the convenience. [Left: In the case that 1− ε2c2 > 0.] [Right: In the case that 1− ε2c2 < 0.] Remark 2.9. Note that the families of functions satisfying the equations obtained in Theorem 2.7 and Theorem 2.8 are lumped together in a rough form in Figure 2, although their asymptotic behavior is strictly different. Theorem 2.10. Assume that α ∈ 2N, ε > 0, and 1− ε2c2 > 0. Then, for a given positive constant 0 < c < 1/ε, the equation (1.1) has a family of the traveling wave solutions (which corresponds to a family of the orbits of (1.5)) with singularities at ξ → −∞ and ξ → +∞. Moreover, its each function u(t, x) ≡ 1− φ(ξ) satisfies the following: • limξ→+∞ φ(ξ) = +∞, limξ→−∞ φ(ξ) = +∞. 8 Y. ICHIDA EJDE-2023/05 • φ(ξ) > 0 holds for ξ ∈ R. • There exists a constant ξ∗ ∈ R such that the following holds: ψ(ξ) < 0 for ξ ∈ (−∞, ξ∗), ψ(ξ∗) = 0 and ψ(ξ) > 0 for ξ ∈ (ξ∗,+∞). In addition, the asymptotic behavior for ξ → +∞ and ξ → −∞ are φ(ξ) ∼ O(ξ 1 α+1 ), ψ(ξ) ∼ O((−ξ)− α α+1 ), (2.9) as ξ → +∞, and φ(ξ) ∼ Ce− c 1−ε2c2 ξ as ξ → −∞, (2.10) with C > 0. On the other hand, assume that 1−ε2c2 < 0. Then, for a given positive constant c > 1/ε, the equation (1.1) has a family of the traveling wave solutions (which corresponds to a family of the orbits of (1.5)) with singularities at ξ → −∞ and ξ → +∞. Moreover, its each function u(t, x) ≡ 1− φ(ξ) satisfies the following: • limξ→+∞ φ(ξ) = −∞, limξ→−∞ φ(ξ) = −∞. • φ(ξ) < 0 holds for ξ ∈ R. • There exists a constant ξ∗ ∈ R such that the following holds: ψ(ξ) > 0 for ξ ∈ (−∞, ξ∗), ψ(ξ∗) = 0 and ψ(ξ) < 0 for ξ ∈ (ξ∗,+∞). In addition, the asymptotic behavior for ξ → +∞ and ξ → −∞ are φ(ξ) ∼ −Ce− c 1−ε2c2 ξ as ξ → +∞ (2.11) with C > 0, and φ(ξ) ∼ O(ξ 1 α+1 ), ψ(ξ) ∼ O((−ξ)− α α+1 ), (2.12) as ξ → −∞. ξ u = 1− φ u = 1 0 ξ u = 1− φ u = 1 0 Figure 3. Schematic picture of the each traveling wave solutions with the singularities at ξ → −∞ and ξ → +∞ in obtained Theo- rem 2.10. [Left: In the case that 1− ε2c2 > 0.] [Right: In the case that 1− ε2c2 < 0.] Note that in the figure on the right, the trajec- tory in which the maximum of u is above the ξ-axis is chosen from among the infinitely many trajectories that correspond to Theo- rem 2.10. EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 9 Remark 2.11. We mentioned that when 1 − ε2c2 = 0, φ(ξ) can be expressed explicitly as in (1.4). This is the same as the number of order of φ(ξ) in (2.7), (2.8), (2.9), and (2.12). However, the detailed relationships and mathematical meanings of these are not known yet, and will be the subject of future work. Remark 2.12. Some functions obtained in the above Theorems satisfy the equa- tion only on finite interval or semi-infinite interval. In this paper, we do not discuss the behavior of the solutions of (1.5) after ψ(ξ) becomes infinity (outside of the interval on that φ(ξ) satisfies (1.3)). It is necessary that more detailed (and hard) analysis in order to study the solutions after ψ(ξ) reaches the singularity. So we leave it open here. It should be noted that equation (1.1) is invariant under transla- tion for spatial coordinates, so many of the same waves connected together should also satisfy the equation, except at the points where the derivatives diverge. How- ever, since our interest in this paper is to study the traveling waves of (1.5) from the viewpoint of dynamical systems, we do not discuss this paper. 3. Dynamics on the Poincaré disk of (1.5) In this section, we study R2∪{(φ, ψ) | ‖(φ, ψ)‖ = +∞}, i.e., the dynamics on the Poincaré disk, by the Poincaré compactification. In order to study the dynamics of (1.5) on the Poincaré disk, we desingularize it by the time-scale desingularization ds/dξ = φ−α for α ∈ 2N. (3.1) Since that α is even, the direction of the time does not change via this desingular- ization. Then, we have φ′ = φαψ, ψ′ = (1− ε2c2)−1(−cφαψ + 1), (3.2) where ′ = d ds , with 1− ε2c2 6= 0. This system (3.2) does not have equilibria. It should be noted that the time scale desingularization (3.1) is simply multi- plying the vector field by φα. Then, except the singularity {φ = 0}, the solution curves of the system (vector field) remain the same but are parameterized differ- ently. Still, we refer to [13, Section 7.7] and references therein for the analytical treatments of desingularization with the time rescaling. In what follows, we use similar time rescaling (re-parameterization of the solution curves) repeatedly to desingularize the vector fields. Now we can consider the dynamics of (3.2) on the charts U j and V j (j = 1, 2). See [4, 10, 11] and their references for definitions of these local coordinates. Note that these results described below are consistent with the process shown in [10, Theorem 2] and [11, Proposition 1], assuming ε = 0. For the reader’s convenience, the calculation process is described here considering the case where ε > 0. 3.1. Dynamics on the chart U2. To obtain the dynamics on the chart U2, we introduce coordinates (λ, x) by the formulas φ = x/λ, ψ = 1/λ. In this chart, it corresponds to φ → 0 and ψ → +∞ and the direction in which x is positive corresponds to the direction in which φ is positive. See [10, Figure 2] for a geometric image. Then, these transformations yield λ′ = (1− ε2c2)−1(cλ−α+1xα − λ2), 10 Y. ICHIDA EJDE-2023/05 x′ = λ−αxα + (1− ε2c2)−1(cλ−αxα+1 − λx), where ′ = d ds . By using the time-scale desingularization dτ/ds = λ−α, we have λτ = (1− ε2c2)−1(cλxα − λα+2), xτ = xα + (1− ε2c2)−1(cxα+1 − λα+1x), (3.3) where λτ = dλ/dτ and xτ = dx/dτ . System (3.3) has the equilibria E+ 0 : (λ, x) = (0, 0), Ec : (λ, x) = (0,M1), M1 = −(1− ε2c2)c−1. The Jacobian matrices of the vector field (3.3) at these equilibria are E+ 0 : ( 0 0 0 0 ) , Ec : ( M2 0 0 M2 ) , M2 = (1− ε2c2)α−1 cα−1 . Therefore, Ec is a source when 1 − ε2c2 > 0, and a sink when 1 − ε2c2 < 0. The equilibrium E+ 0 is not hyperbolic. Thus, to determine the dynamics near E+ 0 , we desingularize it by introducing the blow-up coordinates λ = rα−1λ̄, x = rα+1x̄ (see [4]). Since we are interested in the dynamics on the Poincaré disk, we consider the dynamics of blow-up vector fields on the charts {λ̄ = 1} and {x̄ = ±1} (see also [10, 11]). 3.1.1. Dynamics on the chart {λ̄ = 1}. By the change of coordinates λ = rα−1, x = rα+1x̄, we have rτ = r(α− 1)−1(1− ε2c2)−1(crα(α+1)x̄α − rα2−1), x̄τ = 2(α− 1)−1(1− ε2c2)−1(rα 2−1x̄− crα(α+1)x̄α+1) + rα 2−1x̄α. The time-rescaling dη/dτ = rα 2−1 yields rη = (α− 1)−1(1− ε2c2)−1(crα+2x̄α − r), x̄η = 2(α− 1)−1(1− ε2c2)−1(x̄− crα+1x̄α+1) + x̄α, (3.4) where rη = dr/dη and x̄η = dx̄/dη. The equilibria of (3.4) on {r = 0} are E + 0 : (r, x̄) = (0, 0), E + α : (r, x̄) = (0,M3), M3 = [−2(α− 1)−1(1− ε2c2)−1] 1 α−1 . Note that M3 < 0 when 1− ε2c2 > 0 and M3 > 0 when 1− ε2c2 < 0. The Jacobian matrices of the vector field (3.4) at these equilibria are E + 0 : ( − 1 (α−1)(1−ε2c2) 0 0 2 (α−1)(1−ε2c2) ) , E + α : (− 1 (α−1)(1−ε2c2) 0 0 − 2 1−ε2c2 ) . Therefore, E + 0 is a saddle, and E + α is a sink in the case that 1 − ε2c2 > 0. In addition, E + 0 is a saddle, and E + α is a source in the case that 1− ε2c2 < 0. Furthermore, since |−(α−1)−1(1−ε2c2)−1| < |−2(1−ε2c2)−1| holds, trajectories near E + α are tangent to {x̄ = M3, r ≥ 0} as η → +∞. The solutions around E + α are approximated as r(η) = C1e − 1 (α−1)(1−ε2c2) η (1 + o(1)), x̄(η) = C2e − 2 1−ε2c2 η +M3(1 + o(1)), (3.5) as η → +∞, with constants C1 and C2. EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 11 3.1.2. Dynamics on the chart {x̄ = 1}. By the change of coordinates λ = rα−1λ̄, x = rα+1, and time-rescaling dη/dτ = rα 2−1, we have rη = (α+ 1)−1r + (α+ 1)−1(1− ε2c2)−1(crα+2 − rλ̄α+1), λ̄η = −(α− 1)(α+ 1)−1λ̄+ (α+ 1)−1(1− ε2c2)−1(2crα+1λ̄− 2λ̄α+2). (3.6) If 1 − ε2c2 > 0, then the equilibrium on {r = 0, λ̄ ≥ 0} is (r, λ̄) = (0, 0). The Jacobian matrix of the vector field (3.6) at this equilibrium is (0, 0) : ( 1 α+1 0 0 −α−1α+1 ) . Therefore, the equilibrium (0, 0) is a saddle. If 1− ε2c2 < 0, system (3.6) has the equilibria on {r = 0, λ̄ ≥ 0} (r, λ̄) = (0, 0), (r, λ̄) = (0,M4), M4 = [−2−1(α− 1)(1− ε2c2)] 1 α+1 > 0. The Jacobian matrices of the vector field (3.6) at these equilibria are (0, 0) : ( 1 α+1 0 0 −α−1α+1 ) , (0,M4) : ( 1 2 0 0 α− 1 ) . Therefore, the equilibrium (0, 0) is a saddle, and (0,M4) is a source. 3.1.3. Dynamics on the chart {x̄ = −1}. The change of coordinates λ = rα−1λ̄, x = −rα+1, and time-rescaling dη/dτ = rα 2−1 yield rη = −(α+ 1)−1r + (α+ 1)−1(1− ε2c2)−1(crα+2 − rλ̄α+1), λ̄η = (α− 1)(α+ 1)−1λ̄+ (α+ 1)−1(1− ε2c2)−1(2crα+1λ̄− 2λ̄α+2). (3.7) If 1− ε2c2 > 0, the system (3.7) has the equilibria on {r = 0, λ̄ ≥ 0} (r, λ̄) = (0, 0), (r, λ̄) = (0,M5), M5 = [2−1(α− 1)(1− ε2c2)] 1 α+1 > 0. The Jacobian matrices of the vector field (3.7) at these equilibria are (0, 0) : (− 1 α+1 0 0 α−1 α+1 ) , (0,M5) : ( − 1 2 0 0 −α− 1 ) . Therefore, the equilibrium (0, 0) is a saddle, and (0,M5) is a sink. If 1 − ε2c2 < 0, then the equilibrium on {r = 0, λ̄ ≥ 0} is (r, λ̄) = (0, 0). Eigenvalues of the linearized matrix are −(α + 1)−1 and (α − 1)(α + 1)−1 with corresponding eigenvectors (1, 0) and (0, 1), respectively. Therefore, the equilibrium (0, 0) is a saddle. Combining the dynamics on the charts {λ̄ = 1} and {x̄ = ±1}, we can obtain the dynamics on U2 (see Figure 4). The figure for the case 1 − ε2c2 < 0 can be drawn in the same way as for the case 1− ε2c2 > 0. 3.2. Dynamics on the chart V 2. In this chart, it corresponds to φ → 0 and ψ → −∞ and the direction in which x is negative corresponds to the direction in which φ is positive. The change of coordinates φ = −x/λ, ψ = −1/λ give the projected dynamics of (3.2) on the chart V 2: λτ = (1− ε2c2)−1(cλxα + λ2+α), xτ = xα + (1− ε2c2)−1(cxα+1 + λα+1x), (3.8) 12 Y. ICHIDA EJDE-2023/05 �✁✂✄☎✆✝✞ ✟✂ ✠✡☛☞✌✍✎ ☛✏ {λ̄ = 1} {x̄ = 1} {x̄ = −1} λ x λ λ x x λ λ x x E+ 0 U2 E+ 0 Figure 4. Schematic pictures of the dynamics of the blow-up vec- tor fields and U2 in the case that 1− ε2c2 > 0. where τ is the new variable introduced by dτ/ds = λ(s)−α. The system (3.8) has the equilibria E−0 : (λ, x) = (0, 0), Ec′ : (λ, x) = (0,M1), M1 = −(1− ε2c2)c−1. The Jacobian matrices of the vector field (3.8) at these equilibria are the same as that of U2. The system (3.8) can be transformed into (3.3) by the change of coordinates (λ, x) 7→ (−λ, x). Therefore, it is sufficient to consider the blow-up of singularity E−0 : (λ, x) = (0, 0) by the formulas λ = rα−1, x = rα+1x̄ with λ̄ = 1. Then, we have rη = (α− 1)−1(1− ε2c2)−1(crα+2x̄α + r), x̄η = 2(α− 1)−1(1− ε2c2)−1(−x̄− crα+1x̄α+1) + x̄α, (3.9) where η satisfies dη/dτ = rα 2−1. The equilibria of (3.9) on {r = 0} are E − 0 : (r, x̄) = (0, 0), E − α : (r, x̄) = (0,M6), M6 = [2(α− 1)−1(1− ε2c2)−1] 1 α−1 . Note that M6 > 0 when 1 − ε2c2 > 0 and M6 < 0 when 1 − ε2c2 < 0. The equilibrium E − 0 is a saddle with the eigenvalues (α− 1)−1(1− ε2c2)−1 and −2(α− 1)−1(1−ε2c2)−1 whose corresponding eigenvectors are (1, 0) and (0, 1), respectively for both 1− ε2c2 > 0 and 1− ε2c2 < 0. EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 13 When 1 − ε2c2 > 0 holds, E − α is a source with the eigenvalues (α − 1)−1(1 − ε2c2)−1 and 2(1 − ε2c2)−1 whose corresponding eigenvectors are (1, 0) and (0, 1), respectively. Furthermore, E − α is a sink in the case that 1− ε2c2 < 0. The solutions around E − α are approximated as r(η) = C1e 1 (α−1)(1−ε2c2) η (1 + o(1)), x̄(η) = C2e 2 1−ε2c2 η +M6(1 + o(1)), as η → −∞, with constants C1 and C2. This equation will be used later in Subsec- tion 4.1 to derive the asymptotic behavior for ξ → ξ− + 0. 3.3. Dynamics on the chart U1. Let us study the dynamics on the chart U1. In this chart, it corresponds to φ→ +∞ and ψ → 0. The transformations φ = 1/λ, ψ = x/λ yield λτ = −λx, xτ = (1− ε2c2)−1(−cx+ λα+1)− x2, (3.10) via time-rescaling dτ/ds = λ−α. This system has the equilibria e+0 : (λ, x) = (0, 0), ec : (λ, x) = (0,M7), M7 = −c(1− ε2c2)−1. Note that M7 > 0 when 1− ε2c2 < 0 and M7 < 0 when 1− ε2c2 > 0. The Jacobian matrices of the vector field (3.10) at these equilibria are e+0 : ( 0 0 0 M7 ) , ec : ( −M7 0 0 −M7 ) . Therefore, ec is a source when 1 − ε2c2 > 0, and should matches Ec′ . When 1− ε2c2 < 0, ec is sink and should matches Ec. In a similar way to [10, 11], the dynamics near e+0 can be determined by the center manifold theorem (for instance, see [3] for the details of it). The approximation of the (graph of) center manifold can be obtained as follows: {(λ, x) | x = λα+1/c+O(λ2α+2)}. Further, we can see that the dynamics of (3.10) near (0, 0) is topologically equivalent to the dynamics of the following equation: λτ = −λα+2/c+O(λ2α+3). These results were also obtained in [10, 11]. However, we reproduce them since they will be used in the proof of Theorem 2.10 later. 3.4. Dynamics on the chart V 1. In this chart, it corresponds to φ → −∞ and ψ → 0. The transformations φ = −1/λ, ψ = −x/λ yield λτ = −λx, xτ = (1− ε2c2)−1(−cx− λα+1)− x2 (3.11) via time-rescaling dτ/ds = λ−α. We can see that the system (3.11) can be trans- formed into the system (3.10) by the change of variables: (λ, x) 7→ (−λ, x). Thus, with the exception of {λ = 0}, the dynamics of (3.11) is immediately obvious from 14 Y. ICHIDA EJDE-2023/05 the dynamics of (3.10), however, we summarize the results for the derivation of the asymptotic behavior (as it is necessary for the proof of Theorem 2.8). This system has the equilibria e−0 : (λ, x) = (0, 0), ec′ : (λ, x) = (0,M7), M7 = −c(1− ε2c2)−1. The Jacobian matrices of the vector field (3.11) at these equilibria are e−0 : ( 0 0 0 M7 ) , ec′ : ( −M7 0 0 −M7 ) . Therefore, ec′ is a source when 1 − ε2c2 > 0, and should matches Ec. When 1 − ε2c2 < 0, ec′ is sink and should matches Ec′ . The dynamics near e−0 can be determined by the center manifold theorem. In the same way as above, the approximation of the (graph of) center manifold can be obtained as {(λ, x) | x = −λα+1/c+O(λ2α+2)}. (3.12) Further, we can see that the dynamics of (3.11) near (0, 0) is topologically equivalent to the dynamics of the equation λτ = λα+2/c+O(λ2α+3). (3.13) 3.5. Dynamics and connecting orbits on the Poincaré disk. Combining the dynamics on the charts U j and V j (j = 1, 2), we obtain the dynamics on the Poincaré disk that is equivalent to the dynamics of (1.5) (or (3.2)) in the case that α is even as the following Proposition (see also Figure 5). We set the phase space Φ as follows: Φ = {(φ, ψ) : (φ, ψ) ∈ R2 ∪ {‖(φ, ψ)‖ = +∞}}. Note that in Figure 5, the circumference corresponds to {‖(φ, ψ)‖ = +∞}. φ ψE+ 0 E− 0 e−0 e+0 Ec Ec′ φ ψE+ 0 E− 0 e−0 e+0 Ec Ec′ Figure 5. Schematic pictures of the dynamics on the Poincaré disk for (1.5) in the case that α ∈ 2N and ε > 0. [Left: Case 1 − ε2c2 > 0.] [Right: Case 1 − ε2c2 < 0.] See also Fig. 4 for the dynamics around E+ 0 for 1− ε2c2 > 0. Proposition 3.1. Assume that α ∈ 2N and ε > 0. Then, the dynamics on the Poincaré disk of the system (1.5) is expressed as Figure 5 in both cases 1−ε2c2 > 0 and 1− ε2c2 < 0. EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 15 Proof. First, the dynamics on the Poincaré disk for the case 1− ε2c2 > 0 is imme- diately shown by the results of [10, 11]. In exactly the same way as [10, 11], it is proved that there exists connecting orbits between E−0 and E+ 0 . Therefore, we can conclude the existence of orbits that connect e−0 and E+ 0 , Ec and E+ 0 , and Ec′ and e+0 . Next, we prove it for the case where 1 − ε2c2 < 0. In (1.5), the transformation of reversing the positive and negative values of 1 − ε2c2 is equivalent to applying the following transformation: φ 7→ −φ, ψ 7→ ψ, ξ 7→ −ξ. (3.14) In fact, (1.5) becomes φ′ = ψ, ψ′ = −(1− ε2c2)−1(−cψ + φ−α) by the transformation in (3.14). This equation corresponds to the reversal of the sign of 1− ε2c2 in (1.5). Thus, the dynamics on the Poincaré disk for 1− ε2c2 < 0 is a symmetry transformation of (3.14) over that for 1− ε2c2 > 0. This completes the proof. � 4. Proofs of main results In this section, we prove the main theorems. If the initial data are located on Φ\{φ = 0}, the existence of the solutions follows from the standard theory for the ordinary differential equations. Therefore, we consider the existence of the trajec- tories that connect equilibria and the detailed dynamics near the equilibria on the Poincaré disk and their asymptotic behavior. The table 1 shows the correspondence between each connecting orbit obtained by the proposition and the traveling wave described in the theorem proved below. Table 1. The correspondence between each connecting orbit obtained by the proposition and the traveling wave described in the theorem proved below. Theorem Connecting orbits Theorem 2.5 between E−0 and E+ 0 Theorem 2.7 between Ec and E+ 0 Theorem 2.8 between e−0 and E+ 0 for 1− ε2c2 > 0 between E+ 0 and e+0 for 1− ε2c2 < 0. Theorem 2.10 between E ′ c and e+0 for 1− ε2c2 > 0 between e−0 and E ′ c for 1− ε2c2 < 0. 4.1. Proof of Theorem 2.5. Proof. The proof of existence of the connecting orbits between E−0 and E+ 0 in both cases 1 − ε2c2 > 0 and 1 − ε2c2 < 0 is obtained in [10, 11], and Proposition 3.1. Therefore, there exists a family of the functions which corresponds to a family of the orbits of (1.5). 16 Y. ICHIDA EJDE-2023/05 Next, we prove the existence of a constant ξ∗ ∈ (ξ−, ξ+). It is sufficient to show the connecting orbits pass through the line {ψ = 0}. This is evident from the existence of connecting orbits in both cases 1 − ε2c2 > 0 and 1 − ε2c2 < 0. Furthermore, this means that we are giving information about the increase or decrease of ψ. Finally, we compute the asymptotic behavior of the trajectories near the equi- libria E−0 and E+ 0 as follows. This derivation is a refinement of the discussion in [10, 11]. Note that the basic idea is the same as the previous ones. However, the detailed principal part is chosen as carefully as in [12] (see Remark 2.4). Assume that 1− ε2c2 > 0. Using (3.5), we then have dη dξ = ds dξ dτ ds dη dτ = φ−αλ−αrα 2−1 = r−α−1x̄−α { C1e − 1 (α−1)(1−ε2c2) η (1 + o(1)) }−α−1 × { C2e − 2 1−ε2c2 η(1 + o(1)) +M3 }−α ∼ C3e α+1 (α−1)(1−ε2c2) η{ C2e − 2 1−ε2c2 η(1 + o(1)) +M3 }−α = C3e α+1 (α−1)(1−ε2c2) η{ C2e − 2 1−ε2c2 η(1 + o(1)) }α + α { C2e − 2 1−ε2c2 η(1 + o(1)) }α−1 M3 + · · ·+ (M3)α ∼ C4e α+1 (α−1)(1−ε2c2) η as η → +∞ with constants Cj . As a note, we emphasize that the last part “∼” corresponds to an improvement from [10, 11] (see Remark 2.4). From this result, we can obtain dξ dη = C5e − α+1 (α−1)(1−ε2c2) η (1 + o(1)) as η → +∞. This yields ξ(η) ∼ C6e − α+1 (α−1)(1−ε2c2) η + C7, C7 ∈ R. Setting ξ+ := limη→+∞ ξ(η), we have ξ+ = ∫ +∞ 0 dξ dη dη = C5 ∫ +∞ 0 e − α+1 (α−1)(1−ε2c2) η dη < +∞. Therefore, ξ+ − ξ ∼ Ce− α+1 (α−1)(1−ε2c2) η as η → +∞ . Finally, we obtain φ(ξ) = x λ = rα+1x̄ rα−1 = r2x = { C1e − 1 (α−1)(1−ε2c2) η (1 + o(1)) }2{ C2e − 2 1−ε2c2 η(1 + o(1)) +M3 } ∼ C8e − 2 (α−1)(1−ε2c2) η{ C2e − 2 1−ε2c2 η(1 + o(1)) +M3 } = C9e − 2α (α−1)(1−ε2c2) η + C8 ·M3e − 2 (α−1)(1−ε2c2) η ∼ −Ce− 2 (α−1)(1−ε2c2) η as η → +∞. EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 17 Note that the process here is also different from the process in [10, 11], as we have chosen a more appropriate principal term. Here, in the last relation, since e − 2α (α−1)(1−ε2c2) η < e − 2 (α−1)(1−ε2c2) η is satisfied by η > 0, we choose the term e − 2 (α−1)(1−ε2c2) η as η → +∞. From the above results, we obtain φ(ξ) ∼ −Ce− 2 (α−1)(1−ε2c2) η ∼ −C(ξ+ − ξ) 2 α+1 as ξ → ξ+ − 0. Since the trajectories are lying on {φ < 0}, it holds that C > 0. Similarly, the asymptotic behavior of ψ(ξ) as ξ → ξ+−0 for 1−ε2c2 > 0 is also derived. Therefore, we can derive (2.1) and (2.2). Furthermore, (2.3) and (2.4) for 1 − ε2c2 < 0 are derived in exactly the same way. This completes the proof. � Remark 4.1. Rewriting the process of deriving the asymptotic behavior in the proof above, we can see that φ′(ξ) ∼ ψ(ξ) as ξ → ξ+ − 0 . This implies that the first equation in (1.5) also holds in the sense of asymptotic behavior. Since this relation does not hold in the results for [10, Theorem 2] and [11, Proposition 1], we believe that this improvement may have improved the accuracy of the asymptotic behavior. 4.2. Proof of Theorem 2.7. Proof. The proof of existence of the connecting orbits between Ec and E+ 0 in both cases 1 − ε2c2 > 0 and 1 − ε2c2 < 0 is obtained in Proposition 3.1. That is, in the same way as in the proof of Theorem 2.5, a family of the functions which corresponds to a family of the orbits of (1.5) is shown. Assume that 1− ε2c2 > 0. In this case, all that remains to be shown is to derive (2.5). The solutions at the around ec′ on the chart V 1 (matches Ec) have the form λτ ∼ C1e −M7τ (1 + o(1)), xτ ∼ C2e −M7τ (1 + o(1)) +M7, M7 = − c 1− ε2c2 , where C1 and C2 are constants. Then dτ dξ = dτ ds ds dξ = λ−αφ−α = 1 . This yields ξ(τ) = τ + C3, (C3 ∈ R). We can see that ξ → −∞ as τ → −∞. This relationship shows that τ(ξ) = ξ + C4, (C3 ∈ R) . Therefore, φ(ξ) = − 1 λ ∼ − { C1e c 1−ε2c2 τ (1 + o(1)) }−1 ∼ −C5e − c 1−ε2c2 τ 18 Y. ICHIDA EJDE-2023/05 = −Ce− c 1−ε2c2 ξ as ξ → −∞ with constants C5 > 0 and C > 0. The reason why C > 0 (C5 > 0) is the trajectories are lying on {φ < 0}. Therefore, (2.5) can be derived. Furthermore, (2.6) for 1 − ε2c2 < 0 is derived in exactly the same way. This completes the proof. � 4.3. Proof of Theorem 2.8. Proof. Assume that 1−ε2c2 > 0. The existence of the orbits connecting e−0 and E+ 0 is as described in Proposition 3.1 above. Note that the same is true for 1− ε2c2 < 0. That is, the existence of the orbits connecting E+ 0 and e+0 is as described in Proposition 3.1 above. As in the previous proofs of the Theorems, this implies the existence of a family of the functions which corresponds to a family of the orbits of (1.5). In this case, all that remains to be shown is to derive (2.7). The proof is almost identical to the proof of Theorem 3 in [11]. However, there are some symbols and parts that are different. We briefly reproduce the proof and describe it below for the reader’s convenience. If the initial value is on the center manifold, the solution at around e−0 on the chart V 1 has the form λ(τ) = α+1 √ 1 −α+1 c τ − (α+ 1) ·A0 = O(τ− 1 α+1 ), x(τ) = 1 (α+ 1)τ + c(α+ 1)A0 +O(λ2α+2) = O(τ−1) as τ → −∞, with a constant A0. These results are derived (3.12) and (3.13). We then have dτ dξ = dτ ds ds dξ = λ−αφ−α = 1. This yields τ(ξ) = ξ + C̃ with a constant C̃. If φ̃(ξ) is a solution of (1.3) (or (1.5)), then φ̃(ξ + θ) is also solution for any θ ∈ R. Therefore, there exists a solution φ(ξ) such that the following holds: φ(ξ) = −λ−1 ∼ O(τ 1 α+1 ) ∼ O(ξ 1 α+1 ) as ξ → −∞. In addition, we can obtain ψ(ξ) = −xλ−1 ∼ −O(ξ 1 α+1 ) ·O(ξ−1) = O(ξ− α α+1 ) as ξ → −∞. Therefore, (2.7) can be derived. Furthermore, (2.8) for 1 − ε2c2 < 0 are derived in exactly the same way. This completes the proof. � 4.4. Proof of Theorem 2.10. Proof. The existence of the orbits connecting Ec′ and e+0 (resp. e−0 ) in the case that 1− ε2c2 > 0 (resp. 1− ε2c2 < 0) is as described in Proposition 3.1. By focusing on Ec′ , (2.10) and (2.11) can be proved in the same way as Theorem 2.7. Furthermore, we assume that 1 − ε2c2 > 0. By focusing on e+0 , (2.9) can be proved in the same way as Theorem 2.8. Similarly, by focusing on e−0 when 1 − ε2c2 < 0, we obtain (2.12). This completes the proof. � EJDE-2023/05 TRAVELING WAVES IN A MEMS TYPE EQUATION 19 5. Concluding remarks The general MEMS type equation is a combination of hyperbolic and parabolic equations as shown in (1.1). In addition, the reaction-diffusion equation with ε = 0 is often considered for convenience of analysis and comparison. In this paper, we studied the existence, information about the shapes, and the asymptotic behavior of traveling waves with the singularity of equation (1.1) by adding ε2utt to the left- hand side of the equation treated in [10]. Furthermore, by reviewing the process of deriving the asymptotic behavior obtained in [10, Theorem 2] and [11, Proposition 1], and by carefully selecting the principal terms, we were able to obtain a better asymptotic behavior than these results (see Remark 4.1). Even if we add ε2utt, the asymptotic behavior obtained by improving the derivation process does not change, and the condition for the wave speed with respect to the shape, which did not appear in [10], is obtained. In other words, the existence of this term and its coefficients have a significant effect on the wave speed and the shapes of the trav- eling waves. These are studied by applying the framework that combines Poincaré compactification, classical dynamical systems theory, and geometric methods for the desingularization of vector fields. Since the addition of this term changes the type of the equation from parabolic to hyperbolic, a rigorous discussion of the mathematical formulation of the solu- tion is necessary. As previously mentioned, since the emphasis of this paper is on discussing how the behavior of traveling waves changes from the viewpoint of dynamical systems, we do not discuss it here and leave it for future work. Acknowledgments. The author was partially supported by JSPS KAKENHI Grant Number JP21J20035. The author would like to express their sincere grat- itude to Professor Takashi Sakamoto (Meiji University) for a lot of helpful and valuable comments. References [1] M. J. Álvarez, A. Ferragut, X. Jarque; A survey on the blow up technique, Int. J. Bifurc. Chaos, 21 (2011), 3108–3118. [2] M. Brunella; Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra, J. Differential Equations, 85 (1990), 338–366. [3] J. Carr; Applications of Center Manifold Theory, Springer, Berlin, 1981. [4] F. Dumortier, J. Llibre, C. J. Artés; Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin Heidelberg, 2006. [5] G. K. Duong, H. Zaag; Profile of a tuchdown solution to a nonlocal MEMS model, Math. Models Appl. Sci., 29 (2019), 1279–1348. [6] R. Folino, C. Lattanzio, C. Mascia; Motion of interfaces for a damped hyperbolic Allen-Cahn equation, Commun. Pure Appl. Anal., 19, No. 9 (2020), 4507–4543. [7] J. S. Guo; Recent developments on a nonlocal problem arising in the micro-electro mechanical system, Tamkang J. Math., 45, No. 3 (2014), 229–241. [8] J. S. Guo, B. C. Huang; Hyperbolic quenching problem with damping in the micro-electro mechanical system device, Discrete Contin. Dyn. System. Ser. B, 19 (2014), 419–434. [9] J. S. Guo, P. Souplet; No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal., 47, No. 1 (2015), 614–625. [10] Y. Ichida, T. O. Sakamoto; Quasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearity, Proc. Japan Acad. Ser. A, 96 (2020), 1–6. [11] Y. Ichida, T. O. Sakamoto; Radial symmetric stationary solutions for a MEMS type reaction- diffusion equation with spatially dependent nonlinearity, Jpn. J. Ind. Appl. Math., 38 (2021), 297–322. 20 Y. ICHIDA EJDE-2023/05 [12] Y. Ichida, T. O. Sakamoto; Stationary solutions for a 1D pde problem with gradient term and negative powers nonlinearity, J. Elliptic Parabol. Equ., 8 (2022), 885–918. [13] C. Kuehn; Multiple Time Scale Dynamics, Springer International Publishing, 2015. [14] K. Matsue; On blow-up solutions of differential equations with Poincaré-type compactifi- caions, SIAM J. Appl. Dyn. Syst., 17 (2018), 2249–2288. [15] K. Matsue; Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs, J. Differential Equations, 267 (2019), 7313–7368. [16] Q. Wang, Estimates for the quenching time of a MEMS equation with fringing field, J. Math. Anal. Appl., 405 (2013), 135–147. Yu Ichida Graduate School of Science and Technology, Meiji University, 1-1-1, Higashimita Tama- ku Kawasaki Kanagawa 214-8571, Japan Email address: ichidayu@meiji.ac.jp 1. Introduction 2. Known and main results 3. Dynamics on the Poincaré disk of (??) 3.1. Dynamics on the chart U2 3.2. Dynamics on the chart V2 3.3. Dynamics on the chart U1 3.4. Dynamics on the chart V1 3.5. Dynamics and connecting orbits on the Poincaré disk 4. Proofs of main results 4.1. Proof of Theorem ?? 4.2. Proof of Theorem ?? 4.3. Proof of Theorem ?? 4.4. Proof of Theorem ?? 5. Concluding remarks Acknowledgments References