Fall detection based on dynamic key points incorporating preposed attention

1.   Introduction

We study the following coupled elliptic system:

where $N\leq 5,$ $\lambda, \mu_1, \mu_2$ are positive constants, $\beta\geq 0$ is a coupling constant. This system is related to the steady state of the following Lotka-Volterra ecological model (please see [22,8,6,5,27,18,21,17,16,11] and the references therein):

with Dirichlet ($u = v = 0, (x,t)\in\partial\Omega\times(0,T)$) or Neumann ($u_{\nu} = v_{\nu} = 0, (x,t)\in\partial\Omega\times(0,T)$) boundary condition, and $\Omega\subset\mathbb{R}^N$ is a $C^2$ domain (possibly unbounded), $u(x,t)$ and $v(x,t)$ represent the population densities of two species respectively, $T\in(0,\infty],$ $d_1, d_2>0,$

The coefficients $a, d$ describe their intrinsic growth rates (positive or not), $b, f\geq0$ represent self-limitation of each species, $c, e$ are the interaction coefficients between the species. In the case $c, e>0$, the system is referred to as a cooperative model; when $c, e<0$, it is a competitive system; when $c<0, e>0,$ it is a predator-prey model. In particular, as $d_1 = d_2 = 1,$ we have

Dancer-Zhang in [8] and Dang-Wang-Zhang in [6] study the dynamics of the competing system in a bounded domain and prove that the solution converges to a stationary point under strong competition for two species and multiple species respectively. Dancer-Wang-Zhang in [5,27] and Zhang in [] study the uniform Hölder continuity for competing species in a bounded domain. In [22] Quittner provides Liouville theorems, universal estimates and periodic solutions for cooperative systems. In [18] Julián López-Gómez et al. describe the coexistence states for the predator-prey model with periodic coefficients and analyzes the dynamics of positive solutions of such models. The steady state system of (3) with Dirichlet boundary condition is

By rescaling, we can readily assume $b = f = 1$, then

When $\Omega$ is bounded, this system has been studied intensively in [4,12,9,13,20] and the references therein. We state some results here. For all cases ($c,e\in\mathbb{R}$), Korman and Leung in [12] give the existence of steady state solutions. For the competing case ($c, e<0$), Cosner and Lazer in [4] investigate the existence, uniqueness, and stability of coexistence states for competing species. In [9] Gui and Lou prove the existence and uniqueness of positive solutions when $\bar{\lambda}_1<a = d<\bar{\lambda}_2$ ($\bar{\lambda}_i$ is the $i_{th}$ eigenvalue of $-\Delta$ in $H^1_0(\Omega)$). In [13] Korman and Leung give some nonexistence results. For cooperative system ($c,e>0$), Korman and Leung in [13] give the necessary and sufficient condition for the existence of a positive solution of (5) when $a>d>\bar{\lambda}_1$, and prove the solution is unique when $c$ or $e$ is small. Lou in [20] shows the nonexistence results of positive solutions when $a<\bar{\lambda}_1,\ d<\bar{\lambda}_1,\ ce\leq1$ and the existence when $a<\bar{\lambda}_1,\ d<\bar{\lambda}_1,\ N\leq5,\ ce>1$, and proves the nonexistence when $a = d<(N-6/N)\bar{\lambda}_1, \ N\geq6$, $\Omega$ is a star-shaped domain and $(e+1)u = (c+1)v$ whenever $a = d$.

In [26] Wei and Yao discuss some results about the corresponding Schrödinger equations in $\mathbb{R}^N$, in [28] Zhang and Wang study the structure of positive solutions to the corresponding Schrödinger system in bounded domains. The main difference between system (1) and the Schrödinger system is that (1) does not have a variational structure. We are going to discuss the uniqueness and the nondegeneracy for positive solutions of (1) in $\mathbb{R}^N$. When $\Omega$ is bounded, people are able to make use of the eigenvalues of $-\Delta$ in $H^1_0(\Omega)$. However, there is only essential spectrum for $-\Delta$ in $\mathbb{R}^N$, so there are more difficulties for (1) in $\mathbb{R}^N$.

In this paper, we first prove the uniqueness of positive solutions for (1), then we show the nondegeneracy of the positive solution and the degeneracy of semi-trivial solutions. We also give a complete classification of positive solutions as $\mu_1 = \mu_2 = \beta.$ By [3], we know any positive solution of (1) is radial and decreases with respect to a point in $\mathbb{R}^N.$ With the help of classical bootstrap argument, the solution $(u,v)$ of (1) is in $C^2(\mathbb{R}^N)\times C^2(\mathbb{R}^N)$ and tends to 0 as $|x|\rightarrow \infty.$

From now on, we assume the solutions of (1) are radial with respect to the origin. We denote the subspace of radial functions of $H^1(\mathbb{R}^N)$ by $H^1_r(\mathbb{R}^N)$.

Our results are dependent on the solution of the following basic equation

By [14], we know (6) has a unique positive solution $w$. Besides, $w$ is radial, decreasing and with the decay rate

for some constant $\alpha_0>0$ (see [10]). We note that the restriction $1\le N\le 5$ guarantees the power $2$ in (6) is subcritical.

Then for $0\le \beta\notin\left[\min\{\mu_1, \mu_2\}, \max\{\mu_1, \mu_2\}\right],$ the system admits a positive solution of the form

If $\mu_1 = \mu_2 = :\mu,$ this simplifies to

In fact, for $N = 1,$ the unique positive solution of (6) is explicitly of the form

Our first result is the following uniqueness property of $(u_0,v_0)$ in $\mathbb{R}^1$.

Theorem 1.1. Assume $N = 1,$ $0\leq\beta\notin\left[\min\{\mu_1, \mu_2\},</italic> <italic> \max\{\mu_1, \mu_2\}\right],$ then $(u_0,v_0)$ in (8) with the exact following form

is the unique positive solution to system (1). Furthermore, both $u_0$ and $v_0$ decay like $\mathrm{e}^{-\sqrt{\lambda}\left|x\right|}$ as $\left|x\right|\rightarrow \infty.$

Remark 1. In fact, if $\min\{\mu_1, \mu_2\}<\beta<\max\{\mu_1, \mu_2\}$, there is no nontrivial solution for system (1). In fact, we multiply the first equation by $v$ and multiply the second equation by $u$ respectively, then integrate over $\mathbb{R}^N$,

Taking the difference, we have

Then if $\min\{\mu_1, \mu_2\}<\beta<\max\{\mu_1, \mu_2\}$, there is no nontrivial solution to system (1).

The second result is the uniqueness of $(u_0,v_0)$ when $\beta>\max\{\mu_1, \mu_2 \}$ in higher dimension.

Theorem 1.2. Assume $\beta>\max\{\mu_1, \mu_2 \},$ $2\leq N\leq 5$, then $(u_0,v_0)$ in (8) is the unique positive solution to system (1). In addition, both $u_0$ and $v_0$ decay like $\mathrm{e}^{-\sqrt{\lambda}\left|x\right|}\left|x\right|^{\frac{1-N}{2}}$ as $\left|x\right|\rightarrow \infty$ by (7).

Remark 2. By [3,Theorem 1], we know any positive solution of (1) is radial and decreases with respect to a point in $\mathbb{R}^N$ ($N\le 5$). What is more, the condition of $N\le 5$ is presumed to guarantee the positive solution of (6) exists and is unique by [14]. In fact, the expression in (8) relies on the positive solution of (6).

Definition 1.3. We say that $(U,V)$ is a nondegenerate solution of system (1) if the solution set of the linearized system

is exactly N-dimensional, specifically,

for some constants $a_j$. If there are other solutions to system (10), we say $(U,V)$ is degenerate.

From the mathematical point of view, the nondegeneracy of positive solutions is an important property for constructing concentrating solutions, that is, we usually use the nondegeneracy to construct single or multiple spike solutions (see [10,26]).

The following theorem is the nondegeneracy property of $(u_0,v_0)$ for system (1).

Theorem 1.4. Assume $\beta\notin\left[\min\{\mu_1,\mu_2\},\max\{\mu_1,\mu_2\}\right],$ then the solution $(u_0,v_0)$ of system (1) in (8) is nondegenerate.

Now we consider the degeneracy of semi-trivial solutions. System (1) has semi-trivial solutions $(\bar{u},0),\ (0,\bar{v}),$ where $\bar{u} = w_1: = \frac{\lambda}{\mu_1}w(\sqrt{\lambda}x),\ \bar{v} = w_2: = \frac{\lambda}{\mu_2}w(\sqrt{\lambda}x),$ $w$ is the unique positive solution of (6) and $\bar{u},\bar{v}$ satisfy

Define

Proposition 1. For the solution $(u,v) = (\bar{u},0)$ as $\beta = a_0$, or $(u,v) = (0,\bar{v})$ as $\beta = b_0$, then the linearized problem of (1) has exactly an one-dimensional set of solutions in $H^1_r(\mathbb{R}^N)$.

The next result brings about the complete classification of all positive solutions of system (1) when $\mu_1 = \mu_2 = \beta.$

Theorem 1.5. Suppose that $N\le5$ and $\mu_1 = \mu_2 = \beta>0,$ then all positive solutions of system (1) have the following form

where $w$ is the unique positive solution of equation (6). Moreover, all positive solutions $(u,v)$ decay like $\mathrm{e}^{-\sqrt{\lambda}\left|x\right|}\left|x\right|^{\frac{1-N}{2}}$ as $\left|x\right|\rightarrow \infty$ by (7). In particular, for $\lambda = \mu_1 = \mu_2 = \beta = 1,$ system (1) has infinitely many positive solutions

2.   Proof of Theorem 1.1

We prove the uniqueness of positive solution for system (1) in one dimension.

Proof of Theorem 1.1. When $N = 1$, wo only need to consider the interval $[0,+\infty)$ due to the symmetry. The system can be simplified as

With the help of variable substitution $(u,v)\rightarrow \left(u\left(\frac{x}{\sqrt{\lambda}}\right), v\left(\frac{x}{\sqrt{\lambda}}\right) \right)$, we assume $\lambda = 1.$ Suppose $(u, \bar{v})$ is a positive solution, then

Set $a = \frac{\beta-\mu_1}{\beta-\mu_2}$ and let $v = \bar{v}/a.$ Then $(u,v)$ solves

Step 1. We multiply the equation of $u$ by $v$, then

In a similar way, we multiply the equation of $v$ by $u$,

Take the difference (12)-(13),

Since $a = \frac{\beta-\mu_1}{\beta-\mu_2},$ the above identity can be reduced to

Integrate (14) over $[0,+\infty)$ and make use of $u^{\prime}(0) = 0 = v^{\prime}(0) = u(+\infty) = v(+\infty)$, we have

When $\mu_1\neq \beta$, from the above identity, we know that if $u\geq v$ or $u\leq v$, then $u\equiv v.$

Step 2. We intend to prove $u-v$ does not change sign.

If not, then $u-v$ changes sign. Set $f = u-v,$ then $f$ satisfies

That is, $f$ satisfies

which is a linear equation for $f$. By the isolated properties of the zeroes for nontrivial solutions of linear ordinary differential equations, $u-v$ cannot equal to 0 in any nonempty interval. Therefore, if $u-v$ changes sign infinitely many times, the zeros of $f$ should tend to $\infty$ (set as +$\infty$ up to a variable change). Assume $x_0>0$ is a zero for a nontrivial solution which is large enough, $f$ satisfies

Since $u(r), v(r)\rightarrow 0 \ \rm{as} \ r\rightarrow +\infty$ and $x_0$ is large, we have $\mu_1u+\mu_2 av-1<0,\ \forall x\in (x_0, +\infty)$. Hence by strong maximum principle, $f$ is equal to 0 in $(x_0, +\infty)$, which is a contradiction with the isolated property of the zeroes. Consequently $u-v$ changes sign only finite times. Thus there exists $r_1$ large enough such that

As a result,

Integrating (14) over $(r_1, +\infty)$, we get

When $\beta>\mu_1$, by (15) and (16), we obtain $(\mu_1-\beta)\int_{r_1}^{+\infty}uv(u-v)<0$ and $-\left(u^{\prime}v-v^{\prime}u\right)(r_1)\leq 0,$ which contradicts with (17). Hence the assumption fails and $u-v$ does not change sign.

When $\beta<\mu_1$, we have $(\mu_1-\beta)\int_{r_1}^{+\infty}uv(u-v)>0,$ and thus $\left(u^{\prime}v-v^{\prime}u\right)(r_1)>0.$

We claim: there exists $r_2>r_1$ such that

If the claim holds, then integrate (14) over $(r_2, +\infty)$ we obtain

which is a contradiction. Hence the assumption fails, and $u-v$ does not change sign.

Step 3. We prove the above claim.

If not, since $\left(u^{\prime}v-v^{\prime}u\right)(r_1)>0$ and $u^{\prime}v-v^{\prime}u$ is continuous, we have

By multiplying the equation of $u$ by $u^{\prime}$, we get

thus

Similarly, multiply the equation of $v$ by $v^{\prime}$,

thus

Take the difference (19)-(20), then

Integrate (21) over $(r_1,+\infty)$, we have

Since $u(r_1) = v(r_1)$, the identity is reduced to

thus we have

Because $0>u^{\prime}(r_1)>v^{\prime}(r_1)$, we have $-\frac12 \left[(u^{\prime})^2-(v^{\prime})^2\right](r_1)>0,$ and by (18) we know $\frac{\beta}{3}\int_{r_1}^{+\infty}(au+v)(u^{\prime}v-v^{\prime}u)\mathrm{d}r>0,$ which contradicts with (22). Therefore, the claim is true.

Finally, we obtain $u = v.$ Then $u$ satisfies

With the substitution $w = (\mu_1+\beta a)u,$ then $w$ solves

Similarly, $v$ solves

With the substitution $w = (\mu_2a+\beta )v,$ then $w$ satisfies

Then by (9), we see $w$ is unique and of the form $w(x) = \frac{6e^x}{\left(1+e^x\right)^2}.$ Hence the unique positive solution of system (1) is

and the decay rate is $\mathrm{e}^{-\sqrt{\lambda}\left|x\right|}$ as $\left|x\right|$ goes to infinity. In this way, we have proved the existence and uniqueness of positive solutions.

3.   Proof of Theorem 1.2

Proof of Theorem 1.2. Firstly, by [3,Theorem 1], we know any positive solution of (1) is radial and decreases with respect to a point in $\mathbb{R}^N$ ($N\le 5$). Moreover, the condition of $N\le 5$ is presumed to guarantee the positive solution of (6) exists and is unique by [14]. In fact, the expression in (8) relies on the positive solution of (6). Also it is straightforward to verify that the $(u_0,v_0)$ in (8) is a positive solution to system (1). From now on, we focus on the uniqueness of $(u_0,v_0).$

Let $(u_1,u_2)$ be a positive solution of system (1) and presume $\beta>\mu_2\ge\mu_1.$ Set $a = \frac{\beta-\mu_1}{\beta-\mu_2}\ge1$ and $\bar{u}_2 = u_2/a,$ then $(u_1,\bar{u}_2)$ solves

Notice that $\mu_2a-\mu_1 = \beta(a-1)>0,$ then

That is,

Define $\Omega_+ = \left\{x\in\mathbb{R}^N\ |\ u_1(x)>\bar{u}_2(x)\right\},$ then $\Omega_+$ is a piece-wise $C^1$ domain. In fact, since $u_1,$ $\bar{u}_2$ are radially non-increasing functions, $u_1-\bar{u}_2$ is radial and with isolated zeroes from the perspective of ODE.

Now multiplying the equation of $u_1$ by $\bar{u}_2$ and integrating over $\Omega_+,$ we get

Multiplying the equation of $\bar{u}_2$ by $u_1$ and integrating over $\Omega_+$ we obtain

With the calculation of (23)-(24),

It can be furthermore reduced to

On the one hand, Since $u_1(x)-\bar{u}_2(x)>0$ in $\Omega_+$ and $u_1(x)-\bar{u}_2(x) = 0$ on $\partial\Omega_+,$ we know

Therefore

On the other hand, because $\mu_1<\beta$ and $u_1(x)-\bar{u}_2(x)>0$ in $\Omega_+,$ we have

Compared with (25), we see that $\Omega_+ = \emptyset.$

Similarly, we can show $\Omega_- = \left\{x\in\mathbb{R}^N\ |\ u_1(x)<\bar{u}_2(x)\right\} = \emptyset.$ Therefore $u_1 = \bar{u}_2.$ Then $u_1$ satisfies

Set $\tilde{u}_1: = u_1\left(\frac{x}{\sqrt{\lambda}}\right),$ then $\tilde{u}_1$ solves

Then $w: = \frac{\mu_1+\beta a}{\lambda}\tilde{u}_1$ solves $\Delta w-w+w^2 = 0,$ which is the unique solution of (6) and decays like $\mathrm{e}^{-\left|x\right|}\left|x\right|^{\frac{1-N}{2}}$ by (7). Therefore

which is the same as the $(u_0,v_0)$ in (8). Furthermore, both $u_1$ and $u_2$ decay like $\mathrm{e}^{-\sqrt{\lambda}\left|x\right|}\left|x\right|^{\frac{1-N}{2}}$ by the above expression and (7). Consequently we have proved the $(u_0,v_0)$ in (8) is the unique positive solution to system (1).

Remark 3. The conclusion does not hold for all $\beta>0$ since there are the examples when $\mu_1 = \mu_2 = \beta$ (see Theorem 1.5).

4.   Proof of Theorem 1.4 and Proposition 1

We shall prove the nondegeneracy of the unique solution of system (1).

Proof of Theorem 1.4. The linearized system for system (1) is

Without loss of generality, we assume $\lambda = 1,$ then $(u_0,v_0) = (c_1w,c_2w): = \left(\frac{\beta-\mu_2}{\beta^2-\mu_1\mu_2}w, \frac{\beta-\mu_1}{\beta^2-\mu_1\mu_2}w\right)$. The linearized system at $(u_0,v_0)$ is reduced to

That is,

With the orthogonal transformation:

then $\Phi_1, \Phi_2$ solve

By [25,Lemma 4.1], the eigenvalues for

are

with eigenspaces

From which we obtain $\Phi_1\in span\left\{\frac{\partial w}{\partial x_j}, j = 1,\cdots,N\right\},$ therefore

Moreover, the condition $\beta\notin\left[\min\{\mu_1,\mu_2\},\max\{\mu_1,\mu_2\}\right]$ implies that

then $\Phi_2 = 0,$ and consequently

Finally, we conclude from (27), (28) and the expression of $(u_0,v_0)$ that the solution set of linearized system (26) is exactly $N$-dimensional, namely,

for some constants $a_j$. Then the positive solution $(u_0,v_0)$ is nondegenerate by Definition 1.3.

We begin to prove the degeneracy of semi-trivial solutions of system (1).

Proof of Proposition 1. 1) Assume $\beta = a_0, (u,v) = (\bar{u},0) = (w_1,0)$, the linearized system is

It is easily seen that $a_0$ can be attained by a radial positive function $\psi_0$ (we assume $\psi_0(0) = 1$). Then by the definition of $a_0,$ there exists a constant $c$ such that $\phi_2 = c\psi_0.$ And via the equation for $\bar{u}$ we know $a_0\leq \mu_1.$ With the substitution

then $(\Phi, \phi_2)$ solves

Set $\tilde{\Phi}(x) = \Phi\left(\frac{x}{\sqrt{\lambda}}\right)$ and make use of $\mu_1\bar{u} = \lambda w(\sqrt{\lambda}x),$ then

Again by [25,Lemma 4.1], we know $\tilde{\Phi} = 0,$ which means $\Phi = 0,$ i.e. $\phi_1 = -\frac{a_0}{2\mu_1-a_0}\phi_2.$ Finally we get the radial solution set of (30) is

Which means the radial solution set of linearized system is one-dimensional. That is, the solution set of the linearized system is at least $N+1$ dimensional. Consequently, the semi-trivial solution $(\bar{u},0)$ is degenerate by Definition 1.3.

2) Assume $\beta = b_0, (u,v) = (0, \bar{v}) = (0,w_2)$, the linearized system is

Similarly, we know $b_0$ can be attained at a radial positive function $\phi_0$ (assume $\phi_0(0) = 1$), and there exists a constant $c$ such that $\phi_1 = c\phi_0.$ Moreover via the equation of $\bar{v}$ we get $b_0\leq \mu_2.$ With the substitution

then $(\phi_1, \Phi)$ solves

Set $\tilde{\Phi}(x) = \Phi\left(\frac{x}{\sqrt{\lambda}}\right)$ and recall $\mu_2\bar{v} = \lambda w(\sqrt{\lambda}x),$ then

Use [25,Lemma 4.1] one more time we know $\tilde{\Phi} = 0,$ so $\Phi = 0,$ which means $\phi_2 = -\frac{b_0}{2\mu_2-b_0}\phi_1.$ Finally we get the radial solution set of (31) is

Hence the radial solution set of linearized system is one-dimensional. Which means the radial solution set of linearized system is one-dimensional. That is, the solution set of the linearized system is at least $N+1$ dimensional. Consequently, the semi-trivial solution $(0,\bar{v})$ is degenerate by Definition 1.3.

5.   Proof of Theorem 1.5

Proof of Theorem 1.5. Firstly, we recall that by [3,Theorem 1] any positive solution of (1) is radial and decreases with respect to a point in $\mathbb{R}^N$ ($N\le 5$). Moreover, the condition of $N\le 5$ is presumed to guarantee the positive solution of (6) exists and is unique by [14]. In fact, the expression in (11) relies on the positive solution of (6). Also it is straightforward to verify that any $(u,v)$ in (11) is a positive solution to system (1) with the assumptions of Theorem 1.5. From now on, we concentrate on proving that any positive solution to system (1) with the assumptions of Theorem 1.5 is of the form (11).

In this case, the solution $(u,v)$ is a radial function and satisfies

Via the transformation

we know that $(\tilde{u},\tilde{v})$ solves

After multiplying the equation for $\tilde{u}$ by $\tilde{v}$, we obtain

Similarly, we have

Take the difference (32)-(33), we get

Integrating over the ball $B_r,$ we see that

which implies $\tilde{u}^{\prime}\tilde{v}-\tilde{v}^{\prime}\tilde{u}\equiv0.$ It readily follows that

Then $\tilde{v}$ satisfies

for some positive constant $C.$ Consequently, $w: = (C+1)\tilde{v}$ satisfies $\Delta w-w+w^2 = 0,$ which is the unique solution of (6) and decays like $\mathrm{e}^{-\left|x\right|}\left|x\right|^{\frac{1-N}{2}}$ by (7). Therefore

which is of the form (11). Further, both $u$ and $v$ decay like $\mathrm{e}^{-\sqrt{\lambda}\left|x\right|}\left|x\right|^{\frac{1-N}{2}}$ by the above expression and (7). Therefore we have finished the classification of all positive solutions to system (1) with the assumptions of Theorem 1.5.

Remark 4. One may study the following interesting coupled elliptic system

where $\lambda_1\neq \lambda_2,$ the uniqueness and the nondegeneracy for positive solutions to system (34) are still open, we will continue to do research on these problems.

Acknowledgments

The authors would like to thank the reviewers for the valuable suggestions and comments to improve the manuscript.