Stationary solutions for a 1D pde problem with gradient term and negative powers nonlinearity

Ichida, Yu; Sakamoto, Takashi

doi:10.1007/s41808-022-00180-x

Cited by 3 publications

(12 citation statements)

References 13 publications

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“…This improvement is described in detail in Subsection 4.1. Furthermore, this improvement 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.…”

Section: Known and Main Resultsmentioning

confidence: 99%

“…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).…”

Section: Theoremmentioning

confidence: 99%

“…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]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity

Yu¹

2023

ejde

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We consider traveling waves with singularities in a damped hyperbolic 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 Poincare compactification, classical dynamical systems theory, and geometric methods for the desingularization of vector fields. We report that the presence of this term causes the shapes to change significantly for sufficiently large wave speeds.

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Section: Known and Main Resultsmentioning

confidence: 99%

“…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).…”

Section: Theoremmentioning

confidence: 99%

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Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity

Yu¹

2023

ejde

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“…Here we note that for the one-dimensional case (2), the results stated in [12] cannot be obtained by applying the computations shown in the later sections of this paper straightforwardly (see [12] for the details). Moreover, for N = 2, the dynamics associated with the radially symmetric stationary problem of ( 1) is not topologically equivalent to the case with N 3.…”

mentioning

confidence: 92%

“…Here, α ∈ 2N, 2 = β ∈ 2N, μ > 0, and δ 0. In [12], the existence, profiles, and asymptotic behaviour of the stationary solutions of (2) are studied by applying Poincaré-Lyapunov compactification and dynamical systems theory. Additionally, we note that the results follow from the analysis of the dynamics at infinity.…”

mentioning

confidence: 99%

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

Ichida

Sakamoto

2022

Nonlinearity

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Radially symmetric stationary solutions for a micro-electro-mechanical system (MEMS) type reaction–diffusion equation with fringing field are considered. This equation arises in the study of the MEMS devices. This paper is devoted to the study of the existence of these solutions, information about their shape, and their asymptotic behaviour. These are studied by applying the framework that combines Poincaré type compactification, classical dynamical systems theory, and geometric methods for desingularization of vector fields called the blow-up technique.

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A characterization of nonnegative stationary solutions for certain 1D degenerate parabolic equations and their applications

Ichida

2024

CPAA

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This paper reports results on the existence and characterization of nonnegative stationary solutions (including the weak sense) of a spatial one-dimensional degenerate parabolic equations. The characterization of stationary solutions given in this paper refers to the enumeration of those that exist and the presentation of solution information such as the profile and asymptotic behavior of each of them. Due to the influence of terms derived from the degeneracy of the equations, it is not easy to classify and characterize the existence of stationary solutions of the equations considered in this paper. These results are obtained by using dynamical systems theory and geometric approaches (in particular, Poincaré compactification). In addition, an application of the results obtained in this paper is given. The results of the characterization of nonnegative weak stationary solutions of the spatial onedimensional porous medium equation with special nonlinear terms are shown. These can be obtained by carefully using transformations known from previous studies.

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Stationary solutions for a 1D pde problem with gradient term and negative powers nonlinearity

Cited by 3 publications

References 13 publications

Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity

Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

A characterization of nonnegative stationary solutions for certain 1D degenerate parabolic equations and their applications

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