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

Abstract

Stationary solutions for the one-dimensional partial differential equation with gradient term and negative powers nonlinearity are considered. This equation is a kind of MEMS equation that has the phenomena of MEMS (micro-electro mechanical system) devices as its background. However, it is not easy to understand the behavior of the solution from the effect of the nonlinear term. Therefore, the purpose of this paper is to investigate the properties of a stationary solution that is a typical solution. That is, we prove the existence of stationary solutions including singularities, and give information about their shapes and the asymptotic behavior. Here, the stationary solution with singularity here means a solution that allows infinity or a solution with an infinite differential coefficient. These are studied by applying the framework that combines the Poincaré–Lyapunov compactification and classical dynamical systems theory. The key to use these methods is to reveal the dynamics including infinity of an ordinary differential equation satisfied by stationary solutions.

Appendix: Overview of the Poincaré type compactification

The Poincaré type compactification is one of the compactifications of the original phase space (the embedding of ${\mathbb {R}}^{n}$ into the unit upper hemisphere of ${\mathbb {R}}^{n+1}$). In the following, the Poincaré type compactification includes both the Poincaré compactification and the Poincaré–Lyapunov compactification. The difference between the two is that the vector field is either homogeneous or quasi-homogeneous, respectively. In this appendix, we briefly introduce the Poincaré compactification and the Poincaré–Lyapunov compactification (especially the directional compactification).

First, we present an overview of the Poincaré compactification applied to homogeneous vector fields. Here Section 2 of [6,7,8] are reproduced. Also, it should be noted that we refer [2] for more details. Let

$$\begin{aligned} X = P(\phi ,\psi ) \dfrac{\partial }{\partial \phi } + Q(\phi ,\psi ) \dfrac{\partial }{\partial \psi } \end{aligned}$$

be a polynomial vector field on ${\mathbb {R}}^{2}$, or in other words

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\phi }} = P(\phi ,\psi ), \\ {\dot{\psi }} = Q(\phi ,\psi ), \end{array}\right. } \end{aligned}$$

where $\dot{~~}$ denotes d/dt, and P, Q are polynomials of arbitrary degree in the variables $\phi$ and $\psi$.

We consider ${\mathbb {R}}^{2}$ as the plane in ${\mathbb {R}}^{3}$ defined by $(y_{1},y_{2},y_{3})=(\phi ,\psi ,1)$. We consider the sphere ${\mathbb {S}}^{2} = \{ y \in {\mathbb {R}}^{3} \mid y_{1}^{2} + y_{2}^{2}+y_{3}^{2}=1\}$ which we call Poincaré sphere. We divide the sphere into

$$\begin{aligned} H_{+} = \{ y \in {\mathbb {S}}^{2} \mid y_{3}>0\}, \quad H_{-} = \{ y \in {\mathbb {S}}^{2} \mid y_{3}<0\} \end{aligned}$$

and

$$\begin{aligned} {\mathbb {S}}^{1} = \{y \in {\mathbb {S}}^{2} \mid y_{3}=0\}. \end{aligned}$$

Let us consider the embedding of vector field X from ${\mathbb {R}}^{2}$ to ${\mathbb {S}}^{2}$ given by

$$\begin{aligned} f^{+}:{\mathbb {R}}^{2} \rightarrow {\mathbb {S}}^{2}, \quad f^{-}:{\mathbb {R}}^{2} \rightarrow {\mathbb {S}}^{2}, \end{aligned}$$

where

$$\begin{aligned} f^{\pm }(\phi ,\psi ):= \pm \left( \dfrac{\phi }{\Delta (\phi ,\psi )},\dfrac{\psi }{\Delta (\phi ,\psi )},\dfrac{1}{\Delta (\phi ,\psi )} \right) \end{aligned}$$

with $\Delta (\phi ,\psi ) = \sqrt{\phi ^{2}+\psi ^{2}+1}$.

Then we consider six local charts on ${\mathbb {S}}^{2}$ given by $U_{k} = \{y \in {\mathbb {S}}^{2} \mid y_{k}>0\}$, $V_{k} = \{y \in {\mathbb {S}}^{2} \mid y_{k}<0\}$ for $k=1,2,3$. Consider the local projection

$$\begin{aligned} g^{+}_{k} : U_{k} \rightarrow {\mathbb {R}}^{2}, \quad g^{-}_{k} : V_{k} \rightarrow {\mathbb {R}}^{2} \end{aligned}$$

defined as

$$\begin{aligned} g^{+}_{k}(y_{1},y_{2},y_{3}) = - g^{-}_{k}(y_{1},y_{2},y_{3}) = \left( \dfrac{y_{m}}{y_{k}},\dfrac{y_{n}}{y_{k}} \right) \end{aligned}$$

for $m<n$ and $m,n \not = k$. The projected vector fields are obtained as the vector fields on the planes

$$\begin{aligned} {\overline{U}}_{k} = \{y \in {\mathbb {R}}^{3} \mid y_{k} = 1\}, \quad {\overline{V}}_{k} = \{y \in {\mathbb {R}}^{3} \mid y_{k} = -1\} \end{aligned}$$

for each local chart $U_{k}$ and $V_{k}$. We denote by $(x,\lambda )$ the value of $g^{\pm }_{k}(y)$ for any k.

For instance, it follows that

$$\begin{aligned} (g^{+}_{2} \circ f^{+})(\phi ,\psi ) = \left( \dfrac{\phi }{\psi },\dfrac{1}{\psi }\right) = (x,\lambda ), \end{aligned}$$

therefore, we can obtain the dynamics on the local chart ${\overline{U}}_{2}$ by the change of variables $\phi = x/\lambda$ and $\psi = 1/\lambda$. The locations of the Poincaré sphere, $(\phi ,\psi )$-plane and ${\overline{U}}_{2}$ are expressed as Fig. 9. Throughout this paper, we follow the notations used here for the Poincaré compactification. It is sufficient to consider the dynamics on $H_{+}\cup {\mathbb {S}}^{1}$, which is called Poincaré disk.

Fig. 9

Locations of the Poincaré sphere and chart ${\overline{U}}_{2}$

Second, we consider the case that a vector field is a quasi-homogeneous. In this case, it should be noted that we choose appropriate compactifications to consider the information about dynamics at infinity. That is, when the vector field is the quasi-homogeneous, the information at infinity may not be reflected correctly in the Poincaré compactification. Then, we introduce the Poincaré–Lyapunov compactification (the directional compactification) that is based on asymptotically quasi-homogeneous vector fields. Then we define a class of vector fields which are quasi-homogeneous near infinity, which is determined by types and orders. In the following, we reproduce the definitions given in [10] as an aid to understanding the methods used in this paper. See [10, 11] for details.

Definition 1

[10, Definition 2.1] Let $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ be a smooth function. Let $\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n} \ge 0$ with $(\alpha _{1}, \alpha _{2}, \ldots \alpha _{n})\ne (0,0, \ldots , 0)$ be integers and $k \ge 1$. We say that f is a quasi-homogeneous function of type $(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$ and order k if

$$\begin{aligned} f(R^{\alpha _{1}}x_{1}, R^{\alpha _{2}}x_{2}, \ldots , R^{\alpha _{n}}x_{n})=R^{k}f(x_{1}, x_{2}, \ldots , x_{n}),\quad \forall x\in {\mathbb {R}}^{n},\quad R\in {\mathbb {R}}. \end{aligned}$$

Next, let

$$\begin{aligned} X=\sum _{j=1}^{n}f_{j}(x)\dfrac{\partial }{\partial x_{j}} \iff X:\left( \begin{array}{l} f_{1}(x_{1},x_{2}, \ldots , x_{n}) \\ f_{2}(x_{1},x_{2}, \ldots , x_{n}) \\ \quad \quad \quad \vdots \\ f_{n}(x_{1}, x_{2}, \ldots , x_{n}) \end{array} \right) \end{aligned}$$

be a smooth vector field. We say that X, or $f=(f_{1}, f_{2}, \ldots , f_{n})$ is a quasi-homogeneous vector field of type $(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$ and order $k+1$ if each component $f_{j}$ is a quasi-homogeneous function of type $(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n})$ and order $k+\alpha _{j}$.

For applications to general vector fields, we define the following notion.

Definition 2

[10, Definition 2.2] Let $\alpha =(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n})$ be a set of nonnegative integers. Let the index set $I_{\alpha }$ as

$$\begin{aligned} I_{\alpha }:=\{ i\in \{1,2,\ldots , n\} \mid \alpha _{i}>0 \}, \end{aligned}$$

which we shall call the set of homogeneity indices associated with $\alpha =(\alpha _{1}, \alpha _{2},\ldots , \alpha _{n})$. Let $U\subset {\mathbb {R}}^{n}$. We say the domain $U\subset {\mathbb {R}}^{n}$ is admissible with respect to the sequence $\alpha$ if

$$\begin{aligned} U:=\{ x=(x_{1}, x_{2}, \ldots , x_{n})\in {\mathbb {R}}^{n} \mid x_{i}\in {\mathbb {R}}, \,\,\text{ if }\,\, i\in I_{\alpha },\, (x_{j_{1}}, x_{j_{2}}, \ldots , x_{j_{n-l}})\in {\tilde{U}} \}, \end{aligned}$$

where $\{j_{1}, j_{2}, \ldots , j_{n-l}\}=\{1,2, \ldots , n\} \backslash I_{\alpha }$ and ${\tilde{U}}$ is an $(n-l)$-dimensional open set.

Assumptions in Definition 1 indicate $I_{\alpha }\ne \emptyset$. The notion of asymptotic quasi-homogeneity defined below provides a systematic validity of scalings at infinity in many practical applications.

Definition 3

[10, Definition 2.3] Let $f=(f_{1}, f_{2}, \ldots , f_{n}): U\rightarrow {\mathbb {R}}^{n}$ be a smooth function with an admissible domain $U\subset {\mathbb {R}}^{n}$ with respect to $\alpha$ such that f is uniformly bounded for each $x_{i}$ with $i\in I_{\alpha }$, where $I_{\alpha }$ is the set of homogeneity indices associated with $\alpha$. We say that

$$\begin{aligned} X=\sum _{j=1}^{n}f_{j}(x)\dfrac{\partial }{\partial x_{j}} \iff X:\left( \begin{array}{l} f_{1}(x_{1},x_{2}, \ldots , x_{n}) \\ f_{2}(x_{1},x_{2}, \ldots , x_{n}) \\ \quad \quad \quad \vdots \\ f_{n}(x_{1}, x_{2}, \ldots , x_{n}) \end{array} \right) \end{aligned}$$

or simply f is an asymptotically quasi-homogeneous vector field of type $(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{n})$ and order $k+1$ at infinity if

$$\begin{aligned} \lim _{R\rightarrow +\infty }R^{-(k+\alpha _{j})} \bigl \{ f_{j}(R^{\alpha _{1}}x_{1}, R^{\alpha _{2}}x_{2}, \ldots , R^{\alpha _{n}}x_{n}) -R^{k+\alpha _{j}}(f_{\alpha ,k})_{j}(x_{1}, x_{2}, \ldots x_{n}) \bigm \}=0 \end{aligned}$$

holds for any $(x_{1}, x_{2}, \ldots , x_{n})\in U_{1}$, where $f_{\alpha ,k}=((f_{\alpha ,k})_{1}, (f_{\alpha ,k})_{2}, \ldots , (f_{\alpha ,k})_{n}) : U\rightarrow {\mathbb {R}}^{n}$ is a quasi-homogeneous vector field of type $(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})$ and order $k+1$, and

$$\begin{aligned} U_{1}:=\{ x=(x_{1}, x_{2}, \ldots , x_{n})\in {\mathbb {R}}^{n} \mid (x_{i_{1}}, x_{i_{2}}, \ldots , x_{i_{l}})\in {\mathbb {S}}^{l-1},\, (x_{j_{1}}, x_{j_{2}}, \ldots , x_{j_{n-l}})\in {\tilde{U}} \}, \end{aligned}$$

where $\{i_{1},i_{2},\ldots , i_{l} \}=I_{\alpha }$.

The geometric image of the locational relationship between the Poincaré–Lyapunov sphere corresponding to the Poincaré sphere and the local coordinate ${\overline{U}}_{2}$ is the same as in Fig. 9. Using the type defined in Definition 3 in the case that $n=2$, we consider the dynamics on the local chart ${\overline{U}}_{2}$ by the change of variables $\phi =x/ \lambda ^{\alpha _{1}}$, $\psi =1/\lambda ^{\alpha _{2}}$.