A new Monte Carlo sampling method based on Gaussian Mixture Model for imbalanced data classification

1.   Introduction

In this paper, we study the pointwise long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms

in multi-dimensional half space $\mathbb{R}^n_+: = \mathbb{R}_+\times \mathbb{R}^{n-1}$, with absorbing and radiative boundary condition

${\bf{x}} = (x_1, {\bf{x}}')$ is the space variable with $x_1\in \mathbb{R}_+: = (0,\infty)$, ${\bf{x'}} = (x_2,\cdots, x_n)\in\mathbb{R}^{n-1}$, $t>0$ is the time variable. $\nu_1$ and $\nu_2$ are positive constant viscosities, $a_1$ and $a_2$ are constants. The Laplacian $\Delta = \sum_{j = 1}^n\partial_{x_j}^2$, $f(u)$ is the smooth nonlinear term and $f(u) = O(|u|^k)$ when $k>0$.

Over the past few decades, many mathematicians have concentrated on solving different kinds of damped nonlinear wave equations. The first kind is called the frictional damped wave equation, which is given as follows

see [9,19,20,23] for the references. It is showed that for the long time, the fundamental solution for the linear system of (3) behaves like the Gauss kernel $\frac{e^{-\frac{|{\bf{x}}|^2}{C(t+1)}}}{(t+1)^{\frac{n}{2}}}$. The second kind is called the visco-elastic damped wave, which is given by the following

One can refer to [22] for the decaying rate of the linear solution, [11,12] for the asymptotic profiles of the linear problem, [4,21] for the nonlinear equation, etc. In [9], the authors studied the fundamental solution for the linear system of (4). The results show that the hyperbolic wave transport mechanism and the visco-elastic damped mechanism interact with each other so that the solution behaves like the convented heat kernel, i.e., $\frac{e^{-\frac{(|{\bf{x}}|-ct)^2}{C(t+1)}}}{(t+1)^{\frac{3n-3}{4}}}$ for the odd dimensional case and $\frac{e^{-\frac{(|{\bf{x}}|-ct)^2}{C(t+1)}}}{(t+1)^{\frac{3n-1}{4}}}+\frac{H(ct-|{\bf{x}}|)}{(1+t)^{\frac{3n-2}{4}}(ct-|{\bf{x}}|+\sqrt{t})^{\frac{1}{2}}}$ for the even dimensional case. The solution exhibits the generalized Huygens principle. For other damped wave equations, one can refer to [2,27] for the damped abstract wave equation, and [14,15,16] for the existence and large time behavior of the solutions for the Cauchy problem of mixed damping (both frictional and visco-elastic damping terms are involved) wave equation.

For the initial-boundary value problem of the different damped wave equations, many authors studied the global well-posedness existence, long time behaviors, global attractors and decaying rate estimates of some elementary wave by using delicate energy estimate method, for example [1,13,25,26,28,29]. In this paper, we will use the pointwise estimate technique to give the long time behavior of the solution for system (1) with boundary condition (2). The main part of this technique is the construction and estimation of the Green's functions for the following linear systems:

The way of estimating the Green's functions $\mathbb{G}_i$ was initiated by [17] and developed by [3,5,6,8,10,18,24] and the reference therein. Following the scheme in [10], we will find the relations between the fundamental solutions for the linear Cauchy problem and Green's functions for the linear half space problem, by comparing their symbols in the transformed tangential-spatial and time variables. Then the Green's functions can be described in terms of the fundamental solutions and the boundary surface operator.

With the help of the accurate expression of Green's functions for the linear half space problem and the Duhamel's principle, we get the pointwise long time behavior for the nonlinear solution $\partial_{{\bf{x}}}^{\alpha}u$, $|\alpha|\le1$. We only treat the case $a_1a_2<0$. The boundary condition of Dirichlet type ($a_1 = 0$) and Neumann type ($a_2 = 0$) are much simpler. For the case of $a_1a_2>0$, the linear problem is unstable. The main results of our paper are given as follows:

Theorem 1.1. Let $n = 2,3$ be the spatial dimension, $k>1+\frac{2}{n}$. Assume the initial data $(u_0({\bf{x}}), u_1({\bf{x}}))\in (H^{l+1}\cap W^{l,1})\times (H^{l}\cap W^{l,1})$, $l\ge[\frac{n}{2}]+2$, and satisfy

$\varepsilon$ sufficiently small, then there exists a unique global classical solution to the problem (1) with the mixed boundary condition (2) while $a_{1}a_{2}\le0$. The solution has the following pointwise estimate:

Moreover, we get the following optimal $L^p(\mathbb{R}^n_+)$ estimate of the solution

Remark 1. We can develop a similar theorem for the case of higher space dimension with a suitable choice of $k$ which guarantees the existence of the solution. In Section 2.2, the approximation used in the calculation of the singular part depends on the space dimension. One could modify the short wave part expression of Green's functions for the linear half space problem to prove the results for the general case.

Notations. Let $C$ and $O(1)$ be denoted as generic positive constants. For multi-indices $\alpha = (\alpha_1, \cdots, \alpha_n)$, $\partial_{{\bf{x}}}^\alpha = \partial_{x_1}^{\alpha_1}\cdots\partial_{x_n}^{\alpha_n}$, $|\alpha| = \sum_{i = 1}^n\alpha_i$. Let $L^p$ denotes the usual $L^p$ space on ${\bf{x}}\in\mathbb{R}^n_+$. For nonnegative integer $l$, we denote by $W^{l,p} (1\le p<\infty)$ the usual $L^p-$ Soblev space of order $l$: $W^{l,p} = \{u\in L^p:\partial_{{\bf{x}}}^{\alpha}u\in L^p(|\alpha|\le l)\}(l\ge 1), W^{0,p} = L^p.$ The norm is denoted by $\|\cdot\|_{W^{l,p}} =$ $\|u\|_{W^{l,p}} = \sum_{|\alpha|\le l}\| \partial_{{\bf{x}}}^{\alpha}u \|_{L^p}$. When $p = 2$, we define $W^{l,2} = H^l$ for all $l\ge 0$. We denote $\mathcal{D}_{\delta}: = \{{\boldsymbol{\xi}}\in \mathbb{C}^n| |Im(\xi_i)|\le\delta, i = 1,2,\cdots,n\}$. Introduce the Fourier transform and Laplace transform of $f({\bf{x}},t)$ as follows:

The rest of paper is arranged as follows: in Section 2, we study the fundamental solutions for the linear Cauchy problem and give a pointwise description of the fundamental solutions in $({\bf{x}},t)$ variables. We also describe the fundamental solutions in other transformed variables. In Section 3, the Green's functions for the half space problem are constructed in the transformed tangential-spatial and time domain. By comparing the symbols in the transformed space, we get the relationship between the fundamental solutions and the Green's functions. Finally in Section 4, we give the long time behavior of the solution for the nonlinear problem. Some useful lemmas are given in Appendix.

2.   Fundamental solutions for the linear Cauchy problem

2.1.   Fundamental solutions in $({\boldsymbol{\xi}},t)$ variables

The fundamental solutions for the linear damped wave equations are defined by

Applying the Fourier transform to (7) and (8) in the space variable ${\bf{x}}$, one can compute the fundamental solutions $G_i({\boldsymbol{\xi}},t)$ ($i = 1,2$) in the Fourier space,

2.2.   Fundamental solutions in $({\bf{x}},t)$ variables

In [16], authors have studied the pointwise estimates of the fundamental solutions by long wave-short wave decomposition in the Fourier space. Here we will use the local analysis and inverse Fourier transform to get the pointwise structures of the fundamental solutions in the physical variables $({\bf{x}},t)$. Outside the finite Mach number region $|{\bf{x}}|\ge 3(t+1)$, one can use the weighted energy estimates to get the exponentially decaying estimates of solution in time and space. Inside the finite Mach number region $|{\bf{x}}|\le 4(t+1)$, we will use the long wave short wave decomposition to get the long wave regular parts and short wave singular parts. Here the long wave and short wave are defined as follows:

with the parameter $\varepsilon_0\ll1$, the Heaviside function $H(x)$ is defined by

Long wave component. When $|{\boldsymbol{\xi}}|\le \varepsilon_0\ll 1$, we have the following Taylor expansion for $\sigma_{\pm}$ and $\sigma_+-\sigma_-$:

Then

So we can approximate the fundamental solutions as follows

Using Lemma 5.1 in Appendix, for $|\alpha|\ge0$ we have

Short wave component. We adopt the local analysis method to give a description about all types of singular functions for the short wave component of the fundamental solutions. When $|{\boldsymbol{\xi}}|\ge N$ for $N$ sufficiently large, we have the following Taylor expansion for $\sigma_{\pm}$:

This non-decaying property results in the singularities of the fundamental solution $G_i$ in spatial variable. To investigate the singularities, we approximate the spectra $\sigma_{\pm}$ by $\sigma_{\pm}^*$:

Therefore, the approximated analytic spectra $\sigma_{\pm}^*$ given above satisfy

By Lemma 5.4 in the Appendix, we have

which asserts that all singularities are contained in $\frac{\sigma_+^*e^{\sigma_-^*t}-\sigma_-^*e^{\sigma_+^*t}}{\sigma_+^*-\sigma_-^*}$, $\frac{e^{\sigma_+^*t}-e^{\sigma_-^*t}}{\sigma_+^*-\sigma_-^*}$. Moreover, one can also prove that the errors of this approximation decay exponentially fast in the space-time domain, just like the proof in [7].

Now we seek out all the singularities. For the short wave part of $G_1({\boldsymbol{\xi}},t)$, one breaks

The first term is

It can be estimated as follows

The second term contains no singularities and we have

so

For the third term, the function $\mathcal{F}^{-1} \left[\frac{\sigma_+^*e^{\sigma_-^*t}}{\sigma_+^*-\sigma_-^*}\right]$ does not contain singularities in ${\bf{x}}$ variable due to its asymptotic when $|{\boldsymbol{\xi}}|\rightarrow\infty$ for $t > 0$ :

$K_0 > 0$ and $J_0^*$ is a constant. One has that there exist generic constant $C>0$ such that for $\delta = (-\varepsilon_0,\varepsilon_0)$,

where

We denote

following the way of proof for Lemma 5.4, we get

from (9). So the following estimate for $G_1^S({\bf{x}},t)$ hold,

For the short wave part of $G_2({\boldsymbol{\xi}},t)$, one breaks

The first term is

and we have

The second term contains no singularities. If denoting

then there exists $C>0$ such that

and we have the following estimate for $G_2^S({\bf{x}},t)$,

Hence the short wave components have the following estimates in the finite Mach number region $|{\bf{x}}|\le 4(t+1)$:

Outside the finite Mach number region $|{\bf{x}}|\ge 3(t+1)$.

We choose the weighted function $w$ to be $w = e^{(|{\bf{x}}|-at)/M}$, $M$ and $a$ will be determined later. It satisfies

Consider the linear damped wave equation outside the finite Mach number region:

Denote the outside finite Mach number region $\{{\bf{x}}\in\mathbb{R}^n, |{\bf{x}}|\ge 3(t+1)\}$ by $D_t$ and its boundary by $\partial D_t$. Multiplying each side of the equation in $(10)_1$ by $wu_t$ and integrating with respective to ${\bf{x}}$ on $D_t$, choosing $2<a<3$, $M$ sufficiently large such that $\nu_1>\frac{c^2}{M}$ and $\frac{\nu_1}{2}+\frac{a}{2M} -\frac{\nu_2}{2M^2}>0$, we have

On the boundary $\partial D_t$, by the structures of the fundamental solutions in the finite Mach number region $|{\bf{x}}|\le 4(t+1)$, we have

So

$\delta_0 = \min\{\frac{a}{4M},\frac{\nu_1}{2}+\frac{a}{2M} -\frac{\nu_2}{2M^2}\}$.

One can also get similar estimates for any higher order derivatives $l$:

Integrating (11) and (12) over $t$, using Sobolev's inequality, we have

since $|{\bf{x}}|\ge3(t+1)$. This means that the fundamental solutions $G_i(i = 1,2)$ satisfy the following estimate outside the finite Mach number region $D_t$:

To summarize, we have the following pointwise estimates for the fundamental solutions:

Lemma 2.1. The fundamental solutions have the following estimates for all ${\bf{x}}\in\mathbb{R}^n$, $|\alpha|\ge0$:

Here

$Bessel K_0(|{\bf{x}}|)$ is the modified Bessel function of the second kind with degree $0$.

2.3.   Fundamental solutions in $(x_1,{\boldsymbol{\xi}}',s)$ variables

Applying Laplace transform in $t$ and Fourier transform in ${\bf{x}}$ to the equations in (7) and (8), denoting the transformed variables by $s$ and ${\boldsymbol{\xi}}$ respectively, we get the transformed fundamental solutions in $({\boldsymbol{\xi}},s)$ variables:

Now we give a lemma:

Lemma 2.2.

where $\lambda = \lambda\left({\boldsymbol{\xi}}',s\right) = \frac{\sqrt{(\nu_2s+c^2)|{\boldsymbol{\xi}}'|^2+s^2+\nu_1s}}{\nu_2s+c^2}$.

Proof. We prove it by using the contour integral and the residue theorem. Note that

Define a closed path $\gamma$ containing $\Gamma': = [-R,R]$ while $R$ is a positive constant, $\Omega = \gamma-\Gamma' = \{z|z = Re^{i\theta}\}$.

If $x_1>0$, set $0\le\theta\le\pi$, $R$ is chosen to be sufficiently large such that $\lambda i$ is contained in the domain surrounded by $\gamma$. Consider the contour integral over path $\gamma$. The contribution of the integration over $\Omega$ approaches to $0$ when $R\to\infty$, therefore by the residue theorem, we have for $x_1>0$,

The computation for the case $x_1<0$ is similar. Set $\pi\le\theta\le2\pi$,

Hence we prove this lemma.

With the help of Lemma 2.2, we get the expression of fundamental solutions $G_1$ and $G_2$ in $(x_1,{\boldsymbol{\xi}}',s)$ variables:

In particular, when $\bar{x}_1>0$, we have

3.   The Green's functions for the initial boundary value problem

In this section, we will give the pointwise estimates of the Green's functions for the initial boundary value problem. Firstly, we compute the transformed Green's functions in the partial-Fourier and Laplace transformed space. Then by comparing the symbols of the fundamental solutions and the Green's functions in this transformed space, we get the simplified expressions of Green's functions for the initial-boundary value problem. With the help of the pointwise estimates of the fundamental solutions and boundary operator, we finally get the sharp estimates of Green functions for the half space linear problem.

Before computing, we make the initial value zero by considering the error function $R_i(x_1,{\bf{x}}',t;y_1) = \mathbb{G}_i(x_1,{\bf{x}}',t;y_1)-G_i(x_1-y_1,{\bf{x}}',t)$, which satisfies the following system:

Taking Fourier transform only with respect to the tangential spatial variable ${\bf{x'}}$, Laplace transform with respect to time variable $t$, the following ODE system can be obtained:

Solving it and dropping out the divergent mode as $x_1\to+\infty$, using the boundary relationship, we have

where $\lambda$ is defined in Lemma 2.2.

Therefore the transformed Green's functions $\mathbb{G}_{i}\left(x_1,{\boldsymbol{\xi}}',s;y_1\right)$ $(i = 1,2)$ are

which reveal the connection between fundamental solutions and the Green's functions.

Hence,

Now we estimate the boundary operator $\mathcal{F}^{-1}_{{\boldsymbol{\xi}}'\to {\bf{x}}'}\mathcal{L}^{-1}_{s\to t}\left[\frac{2a_{2}}{a_{2}-a_{1}\lambda}\right]$. The function $\frac{1}{a_{2}-a_{1}\lambda}$ has the poles in the right half time space if $a_1a_2>0$, which suggests that the boundary term will grow exponentially in time. In the following we only consider the case $a_1a_2<0$.

Instead of inverting the boundary symbol, we follow the differential equation method. Notice that

setting

then the function $g\left(x_1, {\bf{x}}',t\right)$ satisfies

Solving this ODE gives

Summarizing previous results we obtain

Lemma 3.1. The Green's functions $\mathbb{G}_i(x_1,{\bf{x}}',t;y_1)$ $(i = 1,2)$ of the linear initial-boundary value problem (5) and (6) can be represented as follows

Meanwhile, the following estimates hold:

and

Proof. Note that

based on the long-wave short-wave decomposition of the fundamental solutions

we can write

and get the estimates directly from Lemma 2.1 and (13).

4.   Long time behavior of solution for the initial-boundary value problem (Proof of theorem 1.1)

The study of boundary operator in the last section suggests that we can only consider the case $a_1a_2<0$ for the nonlinear stability. In [15,16], they proved a threshold $k = 1+\frac{2}{n}$ between global and non-global existence of small data solutions. Here under the assumption of $k>1+\frac{2}{n}$, the global in time existence of solution for the initial-boundary value problem can be proved using the fixed point theorem of Banach, which is similar to the proof given by [16], we omit the details.

Now we give the pointwise long time behavior of the solution for the nonlinear problem and prove the Theorem 1.1. The Green's functions $\mathbb{G}_i(x_1,{\bf{x}}',t;y_1)$($i = 1,2$) give the representation of the solution $u({\bf{x}},t)$:

The initial part $\partial_{{\bf{x}}}^\alpha\mathcal{I}({\bf{x}},t)$ contains two parts:

where

By lemma 5.2, we have the following estimates in the finite Mach number region $|{\bf{x}}|\le 4(t+1)$,

Hence we combine (15) and (16) to get the estimate of the first part in (14) when $|\alpha| = 0$

Similarly, when $|\alpha| = 1$, we have

where

Here we use the integration by parts to estimate the short wave component part. Outside the finite Mach number region, we have

Based on the estimates of (17)-(18), the ansatz is posed for the solution as follows:

Straightforward computations show that

Now we justify the ansatz for the nonlinear term. For $\mathcal N({\bf{x}},t)$, we have

Using Lemma 5.3, one gets

Now we compute the estimate of $\partial_{{\bf{x}}}^\alpha\mathcal{N}$ when $|\alpha| = 1$:

Similarly we have

The boundary term in (19) has the following estimates:

The second term in (19) satisfies

Therefore one has the following estimate for the nonlinear term

Outside the finite Mach number region,

Thus, we verify the ansatz and finish the proof of pointwise estimates of the solution.

The $L^p$ $(p>1)$ estimate can be easily proved using the following equalities:

Hence we finish the proof of Theorem 1.1.

5.   Appendix

Lemma 5.1. [10] In the finite Mach number region $|{\bf{x}}|\le 4(t+1)$, we have the following estimate for the inverse Fourier transform:

Lemma 5.2. [9] We have the follow estimate for $|\alpha|\le 1$ and $r>\frac{n}{2}$,

Lemma 5.3. [9] For ${\bf{x}}\in\mathbb R^n$, $|\alpha|\le 1$, we have

Lemma 5.4. [7] Suppose a function $f\in L^1(\mathbb{R}^n)$ and its Fourier transform $\mathcal{F}[f]({\boldsymbol{\xi}})$ is analytic in $\mathcal{D}_{\delta}$ and satisfies

Then, the function $f({\bf{x}})$ satisfies

for any positive constant $C>1$.

Acknowledgments

The authors would like to thank the referees very much for their valuable comments and suggestions which improve the presentation of papersignicantly.