Global solution and blow-up for the SKT model in Population Dynamics

Ichraf Belkhamsa and Messaoud Souilah

Department of Mathematics, Faculty of Sciencs, Univerity Blida 1, P.O.Box 270,
Blida, Algeria.

ORCID iD: https://orcid.org/0000-0003-0603-5741.

Department of Analysis, Faculty of Mathematics, University of Science and
Technology Houari Boumediene (USTHB), P.O.Box 32, Algiers, Algeria.

ORCID iD: https://orcid.org/0000-0002-2918-3395.

Abstract:

In this paper, we prove the existence and uniqueness of the global solution to
the reaction diffusion system SKT with homogeneous Newmann boundary
conditions. We use the lower and upper solution method and its associated
monotone iterations, where the reaction functions are locally Lipschitz. We
study the blowing-up property of the solution and give sufficient conditions
on the reaction parameters of the model to ensure the blow-up of the solution
in continuous function spaces.

Keywords: Reaction diffusion systems, SKT model, Upper and lower solutions,
Global solution, Blow-up.

##  1 Introduction

Various mathematical models from population dynamics translated into reaction-
diffusion systems posed in a bounded domain of $\mathbb{R}^{n}$. For example,
the Sheguesada Kawasaki Teramoto model (see [28]) proposed in 1978, which
include the following problem:

$\left\\{\begin{array}[]{l}u_{1t}-\Delta[(d_{1}+\alpha_{1}u_{1}+\beta_{1}u_{2})u_{1}]=u_{1}(a_{1}-b_{1}u_{1}+c_{1}u_{2})\
\text{in }Q_{T}\\\
u_{2t}-\Delta[(d_{2}+\alpha_{2}u_{2}+\beta_{2}u_{1})u_{2}]=u_{2}(a_{2}+b_{2}u_{1}-c_{2}u_{2})\
\text{in }Q_{T}\\\ \dfrac{\partial u_{1}}{\partial\eta}=\dfrac{\partial
u_{2}}{\partial\eta}=0\ \text{on }S_{T}=(0,T]\times\partial\Omega\\\
u_{1}(0,x)=u_{1,0}(x),\ u_{2}(0,x)=u_{2,0}(x)\text{ on
}\Omega\end{array}\right.$ (1)

where $\Omega$ is bounded domain in $\mathbb{R}^{n}(n\geq
1),Q_{T}=(0,T]\times\Omega$$,\ S_{T}$ is the boundary and the closure of
$\Omega$. $d_{i},\alpha_{i},\beta_{i},a_{i}$ and $b_{i},c_{i}$ are positive
constants, $\Delta=\sum_{i=1}^{n}\partial^{2}/\partial^{2}x_{i}$ is the
Laplace operator, and $\frac{\partial}{\partial\eta}$ denotes the directional
derivative along the outward normal on $\partial\Omega$. The
problem$(\ref{39})$ has been treated by many researchers, who are devoted to
the blow-up of solution and the global existence with either Newmann or
Dirichle boundary condition by various methods (cf.[29, 7, 8]). In 1984, Kim
[5] proved the universal existence of classical solutions to $(\ref{39})$ for
$n=1$ when $d_{1}=d_{2}$, $\alpha_{1}>0,\beta_{2}>0$, and
$\beta_{1}=\alpha_{1}=0$. Recently, Shim [20] refined Kim’s findings and
established uniform boundaries for solutions by utilizing a new technique. In
several important papers [1], [2], and [3], Amann established the local
existence of a solution to (1). His results were based on the use of the
$W_{p}^{1}$-estimates ($p>1$).

If $di=\alpha_{i}=\beta_{i}=0(i=1,2)$ the problem $(\ref{39})$ is the
historical Volterra model. For $\alpha_{i}=\beta_{i}=0,$ $(\ref{39})$ is the
Lotka-Volterra system; in this case, C.V.Pao in [26] proved that for
$b_{1}c_{2}<c_{1}b_{2}$ the problem admit a unique solution, and for
$b_{1}c_{2}>c_{1}b_{2}$, the solution blow-up. In the same case, Lou, Nagylaki
and Ni [12] studied the effect of diffusion on the blow-up of solution in
finite time $T^{\ast}$. For $n=2$, Y. Lou, Ni, and Wu showed in [14] that the
system (1) has a unique global solution.

On the other hand, Choi, Lui, and Yamada [21] examined the problem when
$\alpha_{2}=0$ and $\beta_{2}=0$, for any value of $n$. However, their
findings demonstrated the global existence of the solution to problem
$(\ref{39})$ only when the coefficient $\beta_{1}$ was sufficiently small.
Subsequently, in 2004, they continued their research and established that the
solution to equation $(\ref{39})$ exists globally either when $\alpha_{2}=0$
or when $\alpha_{2}>0$, with $\beta_{2}=0$ and $n<6$ (See [22]).

In another approach, Linling.Z, Zhi.Li and Zhigui.Li in [16], devoted to the
global existence and blow-up of solutions for system (1) for $\beta_{i}=0$
with Dirichlet boundary condition, the global solution is proved by using the
upper and lower solution method and sufficient conditions are given for the
solution to blow-up. In particular, for $\beta_{i}=0,$ R.Hoyer and M.Souilah
[4] investigated the existence of global and local solution to the reaction-
diffusion system (1), they have given a sufficient condition on the reaction
parameters to ensure the global existence of the solution to the problem in
the Sobolev space $W^{2,p}$.

In this paper, we study the global and blow-up of the solution to the reaction
diffusion system:

$\left\\{\begin{array}[]{l}u_{1t}-\Delta[(d_{1}+\alpha_{1}u_{1})u_{1}]=f_{1}(u_{1},u_{2})\text{\
\ },\text{in }Q_{T}=(0,T]\ast\Omega\\\
u_{2t}-\Delta[(d_{2}+\alpha_{2}u_{2})u_{2}]=f_{2}(u_{1},u_{2}\ ),\text{in
}Q_{T}=(0,T]\ast\Omega\\\ \dfrac{\partial u_{1}}{\partial\eta}=\dfrac{\partial
u_{2}}{\partial\eta}=0,\text{on }S_{T}=(0,T]\times\partial\Omega\\\
u_{1}(0,x)=u_{1,0}(x),\ u_{2}(0,x)=u_{2,0}(x),\text{ on
}\Omega\end{array}\right.$ (2)

where $f_{1}(u_{1},u_{2})=u_{1}(-a_{1}+b_{1}u_{1}-c_{1}u_{2})$,
$f_{2}(u_{1},u_{2})=u_{2}(-a_{2}-b_{2}u_{1}+c_{2}u_{2})$,
$f_{1},f_{2}$ are quasimonotone decreasing.

This paper is arranged as follows: In Section 2, we prove the existence and
uniqueness of a global solution in continuous spaces by using upper and lower
solutions and their associated monotone iterations. Specifically, we create
upper and lower sequences that are monotonous of opposing monotony and
converge to the same limit, the latter is the unique and global solution to
the problem $(\ref{1})$. Thereafter, we give sufficient conditions to ensure
the global existence of $(\ref{1})$. Section 3 proposes a concept based on
creating a lower sequence that reflects the solution to the Cauchy problem,
and we illustrate its convergence towards infinity. Finally, we provide
sufficient conditions to ensure the explosion.

##  2 Existence of global solution

This section aims to establish the existence of a global solution to
$(\ref{1})$. First, we give the definition of ordered upper and lower
solutions of $(\ref{1})$:

###### Definition 1

The pair $w^{(0)}=(w_{1}^{(0)},w_{2}^{(0)}),v^{(0)}=(v_{1}^{(0)},v_{2}^{(0)})$
in $C(\Omega)\cap C^{1,2}(\Omega)$ are called order upper and lower solution
of $(\ref{1}),$ if $w^{(0)}\mathbb{\geq}v^{(0)}$ and $w^{(0)},v^{(0)}$
satisfies the following relations:

$\left\\{\begin{array}[]{l}(w_{1}^{(0)})_{t}-\Delta[(d_{1}+\alpha_{1}w_{1}^{(0)})w_{1}^{(0)}]\mathbb{\geq}w_{1}^{(0)}(-a_{1}+b_{1}w_{1}^{(0)}-c_{1}v_{2}^{(0)})\\\
(w_{2}^{(0)})_{t}-\Delta[(d_{2}+\alpha_{2}w_{2}^{(0)})w_{2}^{(0)}]\mathbb{\geq}w_{2}^{(0)}(-a_{2}-b_{2}v_{1}^{(0)}+c_{2}w_{2}^{(0)})\\\
\dfrac{\partial w_{i}^{(0)}}{\partial\eta}\mathbb{\geq}0\\\
w_{i}^{(0)}(x,0)\mathbb{\geq}u_{i,0}(x)\end{array}\right.$ (3)

and

$\left\\{\begin{array}[]{l}(v_{1}^{(0)})_{t}-\Delta[(d_{1}+\alpha_{1}v_{1}^{(0)})v_{1}^{(0)}]\mathbb{\leq}v_{1}^{(0)}(-a_{1}+b_{1}v_{1}^{(0)}-c_{1}w_{2}^{(0)})\\\
(v_{2}^{(0)})_{t}-\Delta[(d_{2}+\alpha_{2}v_{2}^{(0)})v_{2}^{(0)}]\mathbb{\leq}v_{2}^{(0)}(-a_{2}-b_{2}w_{1}^{(0)}+c_{2}v_{2}^{(0)})\\\
\dfrac{\partial v_{i}^{(0)}}{\partial\eta}\mathbb{\leq}0\\\
v_{i}^{(0)}(x,0)\mathbb{\leq}v_{i,0}(x)\end{array}\right.$ (4)

Let’s define $J\times
J=\\{(u_{1},u_{2})\in(C(\overline{Q}))^{2}:(v_{1},v_{2})\mathbb{\leq}(u_{1},u_{2})\mathbb{\leq}(w_{1},w_{2})\\}$.

We denote $C(\overline{Q}_{T})=C((0,T]\times\overline{\Omega})$ the space of
all bounded and continuous functions in $(0,T]\times\overline{\Omega}$.

Considering only the positive solution of $(\ref{1})$, we define:

$\begin{array}[]{l}P_{i}(u_{i})=h_{i}=\int_{0}^{u_{i}}(d_{i}+2\alpha_{i}s)ds=(d_{i}+\alpha_{i}u_{i})u_{i}\geq
0\\\ h_{it}=(d_{i}+2\alpha_{i}u_{i})u_{it},\ i=1,2\end{array}$ (5)

Where

$u_{i}=\frac{-d_{i}+\sqrt{d_{i}^{2}+4\alpha_{i}h_{i}}}{2\alpha_{i}}=q_{i}(h_{i})$

We may express the problem $(\ref{1})$ in the equivalent form

$\left\\{\begin{array}[]{l}(d_{i}+2\alpha_{i}u_{i})^{-1}h_{it}-\Delta
h_{i}=f_{i}(u_{1},u_{2})\ \text{in }Q_{T}\text{ }\\\ \dfrac{\partial
h_{i}}{\partial\eta}=0\text{ on }S_{T}\\\ h_{i}(x,0)=h_{i,0}(x)\text{ in
}\Omega\\\ u_{i}=q(h_{i})\text{ in }\overline{Q}_{T}\end{array}\right.$ (6)

Define $Lh_{i}=\Delta h_{i}-\varphi_{i}h_{i}$ where $\varphi_{i}>0$ and
$F_{i}(u_{1},u_{2})=f_{i}(u_{1},u_{2})+\varphi_{i}h_{i},\ i=1,2$

For any $u_{1}>0,u_{2}>0$, the problem $(\ref{7})$ is equivalent to

$\left\\{\begin{array}[]{l}(d_{i}+2\alpha_{i}u_{i})^{-1}h_{it}-Lh_{i}=F_{i}(u_{1},u_{2})\text{in
}Q_{T}\text{ }\\\ \dfrac{\partial h_{i}}{\partial\eta}=0\text{ on }S_{T}\\\
h_{i}(x,0)=h_{i,0}(x)\text{ in }\Omega\end{array}\right.$ (7)

First, we present the following positivity lemma:

###### Lemma 1

Let $\sigma(x,t)>0$ in $Q_{T}$, $\beta\geq 0$ on $S_{T}$ and let either

$(i)$ $e(t,x)>0$ in $Q_{T}$ or

$(ii)$ $(-\dfrac{e}{\sigma(x,t)})$ be bounded in $\overline{Q}_{T}.$

If $z\in C^{1,2}(Q_{T})\cap C(\overline{Q}_{T})$ satisfies the following
inequalities:

$\left\\{\begin{array}[]{l}\sigma(t,x)z_{t}-\Delta z+e(x,t)z\geq 0\text{ in
}Q_{T}\text{ }\\\ \dfrac{\partial z}{\partial\eta}+\beta(x,t)z\geq 0\text{ in
}S_{T}\\\ z(x,0)\geq 0\text{ in }\Omega\end{array}\right.$

then $z\geq 0$ in $Q_{T}$

Proof.  See proof the lemma 1.2 of [26].
Using either $w^{(0)}$ or $v^{(0)}$ as the initial iteration. We can construct
a sequence $\left(u^{(k)},h^{(k)}\right)$ from the iteration process for any
$k$ and $i=1,2.$

$\left\\{\begin{array}[]{l}(d_{i}+2\alpha_{i}u_{i}^{(k)})^{-1}h_{it}^{(k)}-Lh_{i}^{(k)}=F_{i}(u_{1}^{(k-1)},u_{2}^{(k-1)})\text{
in }Q_{T}\text{ }\\\ \dfrac{\partial h_{i}^{(k)}}{\partial\eta}=0\text{ on
}S_{T}\\\ h_{i}^{(k)}(x,0)=h_{i,0}^{(k)}(x)\text{ in }\Omega\\\
u_{i}^{(k)}=q(h_{i}^{(k)})\text{ in }\overline{Q}_{T}\end{array}\right.$ (8)

Denote the sequences by $(w^{(k)},\overline{h}^{(k)})$ and
$(v^{(k)},\underline{h}^{(k)})$ respectively, where
$(w^{(0)},\overline{h}^{(0)})$ and $(v^{(0)},\underline{h}^{(0)})$ are the
initial iteration of the sequences.

###  2.1 Monotone sequences

In the following lemma, we show the monotony of the sequences
$(w^{(k)},\overline{h}^{(k)})$ and $(v^{(k)},\underline{h}^{(k)}).$

###### Lemma 2

The sequences $(w^{(k)},\underline{h}^{(k)})$ and
$(v^{(k)},\overline{h}^{(k)})$ defined by $(\ref{9})$ possess the monotone
property

$(v^{(0)},\underline{h}^{(0)})\leq(v^{(k)},\underline{h}^{(k)})\leq(v^{(k+1)},\underline{h}^{(k+1)})\leq(w^{(k+1)},\overline{h}^{(k+1)})\leq(w^{(k)},\overline{h}^{(k)})\leq(w^{(0)},\overline{h}^{(0)})$
(9)

where $\overline{h}$ and $\underline{h}$ are the upper and lower bounds of
$h.$

Proof. $\forall k\geq 1,$ where $\overline{h}=P(w)$ and $\underline{h}=P(v)$

Step 1: Initialisation $(k=1)$

$\displaystyle\text{Let}\quad\underline{z}_{i}^{(1)}=\underline{h}_{i}^{(1)}-\underline{h}_{i}^{(0)}\text{\
and }\overline{z}_{i}^{(1)}=\overline{h}_{i}^{(0)}-\overline{h}_{i}^{(1)}$

for $i=1$, we replace $\underline{z}_{1}^{(1)}$ in $(\ref{9})$

$\displaystyle(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}\underline{z}_{1t}^{(1)}-L_{1}\underline{z}_{1}^{(1)}$
$\displaystyle=$ $\displaystyle
F_{1}(v_{1}^{(0)},w_{2}^{(0)})-(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}\underline{h}_{1t}^{(0)}+L_{1}\underline{h}_{1}^{(0)}$
$\displaystyle=$ $\displaystyle
F_{1}(v_{1}^{(0)},w_{2}^{(0)})+L_{1}\underline{h}_{1}^{(0)}-(d_{1}+2\alpha_{1}v_{1}^{(0)})^{-1}\underline{h}_{1t}^{(0)}+[(d_{1}+2\alpha_{1}v_{1}^{(0)})^{-1}-(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}]\underline{h}_{1t}^{(0)}$
$\displaystyle=$
$\displaystyle\Delta\underline{h}_{1}^{(0)}+f_{1}(v_{1}^{(0)},w_{2}^{(0)})-(d_{1}+2\alpha_{1}v_{1}^{(0)})^{-1}\underline{h}_{1t}^{(0)}+[(d_{1}+2\alpha_{1}v_{1}^{(0)})^{-1}\
-(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}]\underline{h}_{1t}^{(0)}$

We use $(\ref{3})$ to get

$(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}\underline{z}_{1t}^{(1)}-L_{1}\underline{z}_{1}^{(1)}\geq-[(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}-(d_{1}+2\alpha_{1}v_{1}^{(0)})^{-1}]\underline{\
h}_{1t}^{(0)}$

by the mean value theorem, we obtain

$(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}-(d_{1}+2\alpha_{1}v_{1}^{(0)})^{-1}=\frac{-2\alpha_{1}}{(d_{1}+2\alpha_{1}\xi_{1}^{(0)})^{3}}(\underline{h}_{1}^{(1)}-\underline{h}_{1}^{(0)})$

for some intermediate value $\xi_{1}^{(0)}$ between $v_{1}^{(0)}$ and
$v_{1}^{(1)}.$ we have,

$(d_{1}+2\alpha_{1}v_{1}^{(1)})^{-1}\underline{z}_{1t}^{(1)}-L_{1}\underline{z}_{1}^{(1)}+\gamma_{1}^{(0)}\underline{z}_{1}^{(1)}\geq
0\text{ in \ }\Omega\times(0,T],\text{ \ where
}\gamma_{1}^{(0)}=\frac{-2\alpha_{1}}{(d_{1}+2\alpha_{1}\xi_{1}^{(0)})^{3}}\underline{h}_{1t}^{(0)}$

on the other hand,

$\displaystyle\frac{\partial\underline{z}_{1}^{(1)}}{\partial\eta}$
$\displaystyle=$
$\displaystyle\frac{\partial(\underline{h}_{1}^{(1)}-\underline{h}_{1}^{(0)})}{\partial\eta}=\frac{\partial\underline{h}_{1}^{(1)}}{\partial\eta}-\frac{\partial\underline{h}_{1}^{(0)}}{\partial\eta}=0\text{
on }\partial\Omega\times(0,T]$ $\displaystyle\underline{z}_{1}^{(1)}(x,0)$
$\displaystyle=$ $\displaystyle\ \
\underline{h}_{1}^{(1)}(x,0)-\underline{h}_{1}^{(0)}(x,0)\geq 0\text{ in
}\Omega$

It follows from the lemma 1 that $\underline{z}_{1}^{(1)}\geq 0$ and thus
$v_{1}^{(1)}\geq v_{1}^{(0)}.$

We replace $\underline{z}_{2}^{(1)}$ in $(\ref{9})$ and use a same reasoning
we give $\underline{z}_{2}^{(1)}\geq 0$ and thus $v_{2}^{(1)}\geq
v_{2}^{(0)}.$

A similar argument gives $\overline{z}_{i}\geq 0$ and $w_{i}^{(0)}\geq
w_{i}^{(1)}.$
Let $y_{i}^{(1)}=\overline{h}_{i}^{(1)}-\underline{h}_{i}^{(1)},\ i=1,2$

for $i=1$ we know that $y_{1}^{(1)}$ satisfies

$\displaystyle(d_{1}+2\alpha_{1}w_{1}^{(1)})^{-1}y_{1t}^{(1)}-L_{1}y_{1}^{(1)}+\delta_{1}^{(1)}y_{1}^{(1)}=F_{1}(w_{1}^{(0)},v_{2}^{(0)})-F_{1}(v_{1}^{(0)},w_{2}^{(0)})\geq
0$

where
$\delta_{1}^{(1)}=\dfrac{-2\alpha_{1}\underline{h}_{1t}^{(1)}}{(d_{1}+2\alpha_{1}\xi_{1}^{(1)})^{3}}$
with $\xi_{1}^{(1)}$ is the intermediare value between $w_{1}^{(1)}$ and
$v_{1}^{(1)}$

$\begin{array}[]{l}\dfrac{\partial y_{1}^{(1)}}{\partial\eta}=0\\\
y_{1}^{(1)}(x,0)=\overline{h}_{1}^{(1)}(x,0)-\underline{h}_{1}^{(1)}(x,0)=0\end{array}$

It follows again from lemma 1, we have $y_{1}^{(1)}\geq 0,$ implies that$\
\overline{h}_{1}^{(1)}\geq\underline{h}_{1}^{(1)}$ and therefore
$w_{1}^{(1)}\geq v_{1}^{(1)}.$ The same line of reasoning yields
$y_{2}^{(1)}\geq 0,$ which leads to
$\overline{h}_{2}^{(1)}\geq\underline{h}_{2}^{(1)}$and $w_{2}^{(1)}\geq
v_{2}^{(1)\text{ }}.$

The foregoing conclusion shows that

$(v_{i}^{(0)},\underline{h}_{i}^{(0)})\leq(v_{i}^{(1)},\underline{h}_{i}^{(1)})\leq(w_{i}^{(1)},\overline{h}_{i}^{(1)})\leq(w_{i}^{(0)},\overline{h}_{i}^{(0)})$

Step 2 : Iteration $(k>1)$
Assume by induction that

$(v_{i}^{(k)},\underline{h}_{i}^{(k)})\leq(v_{i}^{(k+1)},\underline{h}_{i}^{(k+1)})\leq(w_{i}^{(k+1)},\overline{h}_{i}^{(k+1)})\leq(w_{i}^{(k)},\overline{h}_{i}^{(k)}),k>1$

Let
$\underline{z}_{i}^{(k+1)}=\underline{h}_{i}^{(k+1)}-\underline{h}_{i}^{(k)}$
and
$\overline{z}_{i}^{(k+1)}=\overline{h}_{i}^{(k+1)}-\overline{h}_{i}^{(k)}.$
Using $(\ref{9})$ again, by the quasi monotone nonincreasing of
$(f_{1},f_{2})$, then $\underline{z}_{1}^{(k+1)}$ satisfies

$\left\\{\begin{array}[]{l}(d_{1}+2\alpha_{1}v_{1}^{(k+1)})^{-1}\underline{z}_{1t}^{(k+1)}-L_{1}\underline{z}_{1}^{(k+1)}+\gamma_{1}^{(k)}\underline{z}_{1}^{(k+1)}=F_{1}(v_{1}^{(k)},w_{2}^{(k)})-F_{1}(v_{1}^{(k-1)},w_{2}^{(k-1)})\geq
0\\\ \dfrac{\partial\underline{z}_{1}^{(k+1)}}{\partial\eta}=0\text{ on
}\partial\Omega\times(0,T]\text{ }\\\ \underline{z}_{1}^{(k+1)}(x,0)=0\text{
in }\Omega\end{array}\right.$

where
$\gamma_{1}^{(k)}=\dfrac{-2\alpha_{1}}{(d_{1}+2\alpha_{1}\xi_{1}^{(k)})^{3}}\underline{h}_{1t}^{(k)},\xi_{1}^{(k)}$
is the intermediare value between $v_{1}^{(k)}$ and $v_{1}^{(k+1)}$

by lemma 1, we have $\underline{z}_{1}^{(k+1)}\geq 0$ wich leads to
$\underline{h}_{1}^{(k+1)}\geq\underline{h}_{1}^{(k)}$ and $v_{1}^{(k+1)}\geq$
$v_{1}^{(k)}.$
Similarly, we have $\underline{z}_{2}^{(k+1)}\geq 0$ impilies that
$\underline{h}_{2}^{(k+1)}\geq\underline{h}_{2}^{(k)}$ and $v_{2}^{(k+1)}\geq$
$v_{2}^{(k)}.$ We use the same procedure to get
$\overline{h}_{i}^{(k+1)}\geq\overline{h}_{i}^{(k)}\geq\underline{h}_{i}^{(k+1)},$
and thus $w_{i}^{(k)}\geq w_{i}^{(k+1)}\geq v_{i}^{(k+1)}$ $(i=1,2).$

###  2.2 Convergence of the two sequences

###### Lemma 3

( monotony convergence.) From lemma 2 the sequences
$(w^{(k)},\overline{h}^{(k)})$ and $\left(v^{(k)},\underline{h}^{(k)}\right)$
obtained from $(\ref{9})$ are convergent and verify the following limits

$\lim_{k\rightarrow+\infty}(w^{(k)},\overline{h}^{(k)})=(w,\overline{h}),\text{
}\lim_{k\rightarrow+\infty}(v^{(k)},\underline{h}^{(k)})=(v,\underline{h})$

In the following, we show that
$(w,\overline{h})=(v,\underline{h})=(u^{\ast},h^{\ast})$ where $u^{\ast}$ is
the unique solution for $(\ref{1}).$

###  2.3 Uniqueness of the solution

###### Theorem 4

Let $(w_{1}^{(0)},w_{2}^{(0)}),(v_{1}^{(0)},v_{2}^{(0)})$ be the upper and
lower solution of $(\ref{1}).$ Then the sequences
$(w^{(k)},\overline{h}^{(k)})$ and $(v^{(k)},\underline{h}^{(k)})$ obtained
from $(\ref{9})$ converge monotonically to a unique solution
$(u^{\ast},h^{\ast})$ of $(\ref{7}),$ and they satisfy the relation

$(v^{(0)},\underline{h}^{(0)})\leq(v^{(k)},\underline{h}^{(k)})\leq(v^{(k+1)},\underline{h}^{(k+1)})\leq(u^{\ast},h^{\ast})\leq(w^{(k+1)},\overline{h}^{(k+1)})\leq(w^{(k)},\overline{h}^{(k)})\leq(w^{(0)},\overline{h}^{(0)})\text{,\
}k\geq 1$

The proof of this theorem is similar to Theorem 3.1 in [19].

Creating a pair of ordered upper and lower solutions to $(\ref{1})$ guarantees
a global solution to the problem $(\ref{1})$. In fact, we only need to
construct an upper solution bounded since the lower solution is sufficiently
small and is bounded by $(0,0).$ We have the following theorem

###  2.4 Conditions on the parameters of the model

###### Theorem 5

Let $\lambda_{0}$ be the first eigenvalue of the Laplacian with

$\left\\{\begin{array}[]{l}2\alpha_{1}\lambda_{0}\leq b_{1}\\\
2\alpha_{2}\lambda_{0}\leq c_{2}\\\
(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}>0\end{array}\right.$

If the following conditions hold

$\frac{-a_{1}(2\alpha_{2}\lambda_{0}-c_{2})+a_{2}c_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq\frac{a_{1}c_{2}+a_{2}c_{1}}{c_{2}b_{1}-c_{1}b_{2}}$
$\frac{-a_{2}(2\alpha_{1}\lambda_{0}-b_{1})+a_{1}b_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq\frac{a_{1}b_{2}+a_{2}b_{1}}{c_{2}b_{1}-c_{1}b_{2}}$

then the problem $(\ref{1})$ admits a unique global solution $(u_{1},u_{2})$
in $\overline{\Omega}\times[0,T].$

Proof.  Let $(w_{1}^{(0)},w_{2}^{(0)}),(v_{1}^{(0)},v_{2}^{(0)})$ the upper
and lower solutions of $(\ref{1})$ with

$(w_{1}^{(0)},w_{2}^{(0)})=(N_{1},N_{2}),\quad(v_{1}^{(0)},v_{2}^{(0)})=(v_{1}^{(0)},v_{2}^{(0)})=(\rho_{1}\Phi_{0},\rho_{2}\Phi_{0})$
(10)

where $\Phi_{0}(x)\equiv\Phi_{0}$ is the positive eigenfunction corresponding
to $\lambda_{0}$ satisfying : $\Delta\Phi_{0}=-\lambda_{0}\Phi_{0}$ and
$\dfrac{\partial\Phi_{0}}{\partial\eta}=0.$

$\rho_{i},\ N_{i}$ $($ $i=1,2)$ are some positive constant with $\rho_{i}$
sufficiently small.

The pair $(\ref{32})$ satisfy the inequality $(\ref{2}),(\ref{3})$ if :

$\left\\{\begin{array}[]{l}(N_{1})_{t}-\Delta[(d_{1}+\alpha_{1}N_{1})N_{1}]\mathbf{\geq}N_{1}(-a_{1}+b_{1}N_{1}-c_{1}\rho_{2}\Phi_{0})\\\
(N_{2})_{t}-\Delta[(d_{2}+\alpha_{2}N_{2})N_{2}]\mathbf{\geq}N_{2}(-a_{2}-b_{2}\rho_{1}\Phi_{0}+c_{2}N_{2})\\\
(\rho_{1}\Phi_{0})_{t}-\Delta[(d_{1}+\alpha_{1}\rho_{1}\Phi_{0})\rho_{1}\Phi_{0}]\mathbf{\leq}\rho_{1}\Phi_{0}(-a_{1}+b_{1}\rho_{1}\Phi_{0}-c_{1}N_{2})\\\
(\rho_{2}\Phi_{0})_{t}-\Delta[(d_{2}+\alpha_{2}\rho_{2}\Phi_{0})\rho_{2}\Phi_{0}]\mathbf{\leq}\rho_{2}\Phi_{0}(-a_{2}-b_{2}N_{1}+c_{2}\rho_{2}\Phi_{0})\end{array}\right.$
(11)

from
$\Delta(\Phi_{0}^{2})=2(\Phi_{0}\Delta(\Phi_{0})+\left|\nabla\Phi_{0}\right|^{2})\geq-2\lambda_{0}\Phi_{0}^{2}$
then

$\
\left\\{\begin{array}[]{l}0\mathbf{\geq}-a_{1}+b_{1}N_{1}-c_{1}\rho_{2}\Phi_{0}\\\
0\mathbf{\geq}-a_{2}-b_{2}\rho_{1}\Phi_{0}+c_{2}N_{2}\\\
2\alpha_{1}\rho_{1}\lambda_{0}\Phi_{0}\mathbf{\leq}-a_{1}+b_{1}\rho_{1}\Phi_{0}-c_{1}N_{2}\\\
2\alpha_{2}\rho_{2}\lambda_{0}\Phi_{0}\mathbf{\leq}-a_{2}-b_{2}N_{1}+c_{2}\rho_{2}\Phi_{0}\end{array}\right.$
(12)

As $N_{i}>\rho_{i}\Phi_{0}>0$ the inequalities $(\ref{35})$ become

$\displaystyle-a_{1}+b_{1}N_{1}-c_{1}N_{2}$ $\displaystyle\leq$ $\displaystyle
0$ (13) $\displaystyle-a_{2}-b_{2}N_{1}+c_{2}N_{2}$ $\displaystyle\leq$
$\displaystyle 0$

and

$\displaystyle a_{1}+(2\alpha_{1}\lambda_{0}-b_{1})N_{1}+c_{1}N_{2}$
$\displaystyle\mathbf{\leq}$ $\displaystyle 0$ (14) $\displaystyle
a_{2}+b_{2}N_{1}+(2\alpha_{2}\lambda_{0}-c_{2})N_{2}$ $\displaystyle\leq$
$\displaystyle 0$

with

$2\alpha_{1}\lambda_{0}-b_{1}\leq 0\text{ and
}2\alpha_{2}\lambda_{0}-c_{2}\leq 0$

Assuming that $c_{2}b_{1}>c_{1}b_{2}$, we get from $(\ref{300})$

$N_{1}\leq\frac{a_{1}c_{2}+a_{2}c_{1}}{c_{2}b_{1}-c_{1}b_{2}}\text{ \ \ and \
\ }N_{2}\leq\frac{a_{1}b_{2}+a_{2}b_{1}}{c_{2}b_{1}-c_{1}b_{2}}$

On the outher hand, we find from $(\ref{37})$

$N_{1}\geq\frac{-a_{1}(2\alpha_{2}\lambda_{0}-c_{2})+a_{2}c_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\text{
and \
}N_{2}\geq\frac{-a_{2}(2\alpha_{1}\lambda_{0}-b_{1})+a_{1}b_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\text{\
}$

where

$(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}>0$

The relations $(\ref{300})$ and $(\ref{37})$ are satisfied by any constants
$N_{1},N_{2}$ such that

$\frac{-a_{1}(2\alpha_{2}\lambda_{0}-c_{2})+a_{2}c_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq
N_{1}\leq\frac{a_{1}c_{2}+a_{2}c_{1}}{c_{2}b_{1}-c_{1}b_{2}}$
$\frac{-a_{2}(2\alpha_{1}\lambda_{0}-b_{1})+a_{1}b_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq
N_{2}\leq\frac{a_{1}b_{2}+a_{2}b_{1}}{c_{2}b_{1}-c_{1}b_{2}}$

This suggests that

$\frac{-a_{1}(2\alpha_{2}\lambda_{0}-c_{2})+a_{2}c_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq\frac{a_{1}c_{2}+a_{2}c_{1}}{c_{2}b_{1}-c_{1}b_{2}}$
(15)
$\frac{-a_{2}(2\alpha_{1}\lambda_{0}-b_{1})+a_{1}b_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq\frac{a_{1}b_{2}+a_{2}b_{1}}{c_{2}b_{1}-c_{1}b_{2}}$
(16)

Hence under conditions $(\ref{200})$ and $(\ref{100})$, there exist positive
constants $N_{i},\rho_{i}(i=1,2)$ such that the pair in $(\ref{32})$ are
coupled upper and lower solutions of $(\ref{1})$, we conclude from lemma 3
that the sequences
$\\{w^{(k)},\overline{h}^{(k)}\\},\\{v^{(k)},\underline{h}^{(k)}\\}$ converge
monotonically to somme $(w,\overline{h}),(v,\underline{h}).$ It follows from
theorem 4 that $(w,\overline{h})=(v,\underline{h})\equiv(u^{\ast},h^{\ast})$
is the unique solution of $(\ref{1})$ in $Q_{T}$.

##  3 The Blow-up of the solution

We announce the following general result on blow-up solutions in the partial
differential equations.

###### Theorem 6

Let $(f_{1},f_{2})$ be locally Lipschitz functions in
$\mathbb{R}^{+}\times\mathbb{R}^{+}$ and let $(v_{1}^{(0)},v_{2}^{(0)})$ be a
positive functions defined in $\left[0,T_{0}\right)\times\overline{\Omega}$
for a finite $T_{0}$ and unbounded at some point in $\overline{\Omega}$ as
$t\longrightarrow T_{0}.$ If $(v_{1}^{(0)},v_{2}^{(0)})$ is a lower solution
of $(\ref{1})$ for every $T<T_{0}$ then there exists
$T^{\ast}\mathbf{\leq}T_{0}$ such that a unique positive solution
$\left(u_{1},u_{2}\right)$ to $(\ref{1})$ exists in $\overline{Q}$ and
satisfies the relation :

$\lim_{t\rightarrow T^{\ast}}\max\left[u_{1}(t,x)+u_{2}(t,x)\right]=+\infty$
(17)

The proof of theorem 6 is similar to that of theorem 11.5.1 of [26].
Based on the result of this theorem, the blowing-up property of the solution
to $(\ref{1})$ is ensured if there exists a lower solution unbounded on
$\overline{\Omega}$ at a finite time $T^{\ast}$. The construction of such a
lower solution depends on the type of boundary conditions. For Neumann
boundary conditions, if $(f_{1},f_{2})$ satisfy a growth condition, the lower
solution can often be constructed. We have the growth conditions in the
following lemma.

###  3.1 Conditions on the reaction terms

###### Lemma 7

If $(f_{1},f_{2})$ satisfy the local Lipschitz continuous property and

$\psi_{1}=\min\left\\{\mu_{1}b_{1,}\mu_{2}c_{2}\right\\},\
\psi_{2}=\left(\mu_{1}c_{1}+\mu_{2}b_{2}\right)/2,\text{
}c=\max\left\\{\mu_{1}a_{1},\mu_{2}a_{2}\right\\},\ \psi_{1}>\psi_{2}>0$ (18)

If $\psi=(\psi_{1}-\psi_{2})/2,$ then $(f_{1},f_{2})$ satisfy the growth
condition:

$\mu_{1}f_{1}(u_{1},u_{2})+\mu_{2}f_{2}(u_{1},u_{2})\geq\psi(u_{1}+u_{2})^{2}-c(u_{1}+u_{2}),\
u_{1}\geq 0,\ u_{2}\geq 0$ (19)

where $\mu_{i},\ i=1,2$ are positive constants.

Proof.

$\noindent\mu_{1}f_{1}(u_{1},u_{2})+\mu_{2}f_{2}(u_{1},u_{2})=-(\mu_{1}a_{1}u_{1}+\mu_{2}a_{2}u_{2})+(\mu_{1}b_{1}u_{1}^{2}+\mu_{2}c_{2}u_{2}^{2})-(\mu_{1}c_{1}+\mu_{2}b_{2})u_{1}u_{2}$
(20)

by $(\ref{30})$, the $(\ref{18})$ becomes

$\mu_{1}f_{1}(u_{1},u_{2})+\mu_{2}f_{2}(u_{1},u_{2})=\psi_{1}(u_{1}^{2}+u_{2}^{2})-2\psi_{2}u_{1}u_{2}-c(u_{1}+u_{2})$
(21)

such that

$\psi_{1}(u_{1}^{2}+u_{2}^{2})-2\psi_{2}u_{1}u_{2}=\psi(u_{1}+u_{2})^{2}-\psi(u_{1}^{2}+u_{2}^{2})+\psi_{1}(u_{1}^{2}+u_{2}^{2})-2(\psi+\psi_{1})u_{1}u_{2}$

We get

$\psi_{1}(u_{1}^{2}+u_{2}^{2})-2\psi_{2}u_{1}u_{2}\geq\psi(u_{1}+u_{2})^{2}+2(\psi_{1}-2\psi-\psi_{2})u_{1}u_{2}$

So,

$\psi_{1}(u_{1}^{2}+u_{2}^{2})-2\psi_{2}u_{1}u_{2}\geq\psi(u_{1}+u_{2})^{2}.$

The relation $(\ref{18a})$ becomes

$\mu_{1}f_{1}(u_{1},u_{2})+\mu_{2}f_{2}(u_{1},u_{2})\geq\psi(u_{1}+u_{2})^{2}-c(u_{1}+u_{2}),\
u_{1}\geq 0,\ u_{2}\geq 0$

The lower solution can be constructed by using the solution of the Cauchy
problem:

$\left\\{\begin{array}[]{l}\widehat{p}^{\prime}+\overline{\tau}\widehat{p}=\underline{\psi}\widehat{p}^{2}\\\
\widehat{p}(0)=\widehat{p}_{0}\end{array}\right.$

where $\overline{\tau}$,$\underline{\psi}$ are arbitrary non-zero constants.
In the following, we show the blow-up of the problem $(\ref{1})$

###### Theorem 8

Let $(f_{1},f_{2})$ be locally Lipschitz function in
$\mathbb{R}^{+}\times\mathbb{R}^{+}$ and satisfy the condition $(\ref{30})$
and the relation $(\ref{17})$, and let $\left(u_{1},u_{2}\right)$ be the local
positive solution of $(\ref{1})$. Then for any nonnegative
$\left(u_{1}(0,x),u_{2}(0,x)\right)$ satisfying the condition

$\widehat{p}_{0}>\dfrac{\overline{\tau}}{\underline{\psi}}$ (22)

there exists a finite $T^{\ast}\leq T_{0}$ such that
$\left(u_{1},u_{2}\right)$ possesses the blowing-up property $(\ref{15}),$
where $T_{0}$

$T_{0}=\dfrac{1}{\overline{\tau}}\ln[\dfrac{\underline{\psi}\widehat{p}_{0}}{\underline{\psi}\widehat{p}_{0}-\overline{\tau}}].$
(23)

Proof.  Consider
$(v_{1}^{(0)},v_{2}^{(0)})=(\widehat{p}_{1}(t)\Phi_{0},\widehat{p}_{2}(t)\Phi_{0})$
is a lower solution of $(\ref{1})$ where $\Phi_{0}$ is the positive
eigenfunction corresponding to the eigenvalue $\lambda_{0}$. Let
$\left(u_{1},u_{2}\right)$ be the local positive solution of $(\ref{1})$ and
define the weighted average of the solution $\left(u_{1},u_{2}\right)$ by

$\widehat{p}_{1}(t)=\left|\Omega\right|^{-1}\int\nolimits_{\Omega}\Phi_{0}(x)u_{1}(x,t)dx,\text{
\ \
}\widehat{p}_{2}(t)=\left|\Omega\right|^{-1}\int\nolimits_{\Omega}\Phi_{0}(x)u_{2}(x,t)dx\text{\
and}$
$\widehat{p}(t)=\left|\Omega\right|^{-1}\int\nolimits_{\Omega}\Phi_{0}(x)[\mu_{1}u_{1}(x,t)+\mu_{2}u_{2}(x,t)]dx=\mu_{1}\widehat{p}_{1}(t)+\mu_{2}\widehat{p}_{2}(t)\quad\text{with}\quad\widehat{p}(0)=\widehat{p}_{0}$
(24)

Multiply the first equation in $(\ref{1})$ by $\Phi_{0}$ of the laplacian and
integrate over $\Omega$ using Green’s theorem which implies

$\int_{\Omega}u_{1t}\Phi_{0}dx\geq\int_{\Omega}\Delta\Phi_{0}[(d_{1}+\alpha_{1}u_{1})u_{1}]dx+\int_{\Omega}f_{1}(u_{1},u_{2})\Phi_{0}dx$

$\int_{\Omega}u_{1t}\Phi_{0}dx\geq-\int_{\Omega}\lambda_{0}\Phi_{0}d_{1}u_{1}(x,t)dx-\int_{\Omega}\lambda_{0}\alpha_{1}u_{1}^{2}(x,t)\Phi_{0}dx+\int_{\Omega}f_{1}(u_{1},u_{2})\Phi_{0}dx$
(25)

Multiply $(\ref{22})$ by $\mu_{1}$

$\mu_{1}\left|\Omega\right|\widehat{p}_{1}^{\prime}(t)\geq-\mu_{1}\lambda_{0}d_{1}\left|\Omega\right|\widehat{p}_{1}(t)-\lambda_{0}\alpha_{1}\mu_{1}\left|\Omega\right|\widehat{p}_{1}^{2}(t)+\int_{\Omega}\mu_{1}f_{1}(u_{1},u_{2})\Phi_{0}dx$
(26)

An analogue reasoning for the second equation in $(\ref{1})$, gives

$\mu_{2}\left|\Omega\right|\widehat{p}_{2}^{\prime}(t)\geq-\mu_{2}\lambda_{0}d_{2}\left|\Omega\right|\widehat{p}_{2}(t)-\lambda_{0}\alpha_{2}\mu_{2}\left|\Omega\right|\widehat{p}_{2}^{2}(t)+\int_{\Omega}\mu_{2}f_{2}(u_{1},u_{2})\Phi_{0}dx$
(27)

The combination of the above inequalities $(\ref{23})$ and $(\ref{24})$ gives

$\left|\Omega\right|[\mu_{1}\widehat{p}_{1}^{\prime}(t)+\mu_{2}\widehat{p}_{2}^{\prime}(t)]\geq-\lambda_{0}\left|\Omega\right|[\mu_{1}d_{1}\widehat{p}_{1}(t)+\mu_{2}d_{2}\widehat{p}_{2}(t)]-$
$\lambda_{0}\left|\Omega\right|[\mu_{1}\alpha_{1}\widehat{p}_{1}^{2}(t)+\mu_{2}\alpha_{2}\widehat{p}_{2}^{2}(t)]+\int_{\Omega}[\mu_{1}f_{1}(u_{1},u_{2})+\mu_{2}f_{2}(u_{1},u_{2})]\Phi_{0}dx$
(28)

We notice that

$\widehat{p}^{2}(t)>\mu_{1}^{2}\widehat{p}_{1}^{2}(t)\text{ and
}\widehat{p}^{2}(t)>\mu_{2}^{2}\widehat{p}_{2}^{2}(t)$

If $\lambda_{0}>0$ multiply the two inequalities by
$-\lambda_{0}\left|\Omega\right|\alpha_{1},\
-\lambda_{0}\left|\Omega\right|\alpha_{2}$ respectively, we get

$-\lambda_{0}\left|\Omega\right|\alpha_{1}\widehat{p}^{2}(t)<-\lambda_{0}\left|\Omega\right|\mu_{1}^{2}\alpha_{1}\widehat{p}_{1}^{2}(t)\text{
and
}-\lambda_{0}\left|\Omega\right|\alpha_{2}\widehat{p}^{2}(t)<-\lambda_{0}\left|\Omega\right|\mu_{2}^{2}\alpha_{2}\widehat{p}_{2}^{2}(t)$

taking $\alpha=\alpha_{1}+\alpha_{2},$
$\overline{\mu}=\max\left\\{\mu_{1},\mu_{2}\right\\}$ and substitute in the
above inequalities

$-\lambda_{0}\left|\Omega\right|\alpha\widehat{p}^{2}(t)<-\lambda_{0}\left|\Omega\right|\overline{\mu}[\mu_{1}\alpha_{1}\widehat{p}_{1}^{2}(t)+\mu_{2}\alpha_{2}\widehat{p}_{2}^{2}(t)]$
(29)

for $\overline{d}=\max\left\\{d_{1},d_{2}\right\\}$, then by $(\ref{26})$ and
the growth condition $(\ref{17})$ on $(f_{1},f_{2})$, the relation
$(\ref{25})$ becomes

$\left|\Omega\right|\widehat{p}^{\prime}(t)+\lambda_{0}\left|\Omega\right|\overline{d}\widehat{p}(t)+\lambda_{0}\left|\Omega\right|\alpha\overline{\mu}^{-1}\widehat{p}^{2}(t)>\int_{\Omega}\psi(u_{1}+u_{2})^{2}\Phi_{0}dx-\int_{\Omega}c(u_{1}+u_{2})\Phi_{0}dx$
(30)

We use the relation:

$\overline{\mu}^{-1}(\mu_{1}u_{1}+\mu_{2}u_{2})\leq(u_{1}+u_{2})\leq\underline{\mu}^{-1}(\mu_{1}u_{1}+\mu_{2}u_{2}),\text{
}\underline{\mu}=\min(\mu_{1},\mu_{2})$

We get

$-\int_{\Omega}c(u_{1}+u_{2})\Phi_{0}dx\geq-\int_{\Omega}c\underline{\mu}^{-1}(\mu_{1}u_{1}+\mu_{2}u_{2})\Phi_{0}dx\geq-c\underline{\mu}^{-1}\left|\Omega\right|\widehat{p}(t)$
(31)

and

$\int_{\Omega}\psi(u_{1}+u_{2})^{2}\Phi_{0}dx\geq\psi\overline{\mu}^{-2}\int_{\Omega}(\mu_{1}u_{1}+\mu_{2}u_{2})^{2}\Phi_{0}dx$
(32)

Since by Holder’s inequality,

$\int_{\Omega}\Phi_{0}(\mu_{1}u_{1}+\mu_{2}u_{2})dx\leq(\int_{\Omega}dx)^{1/2}.(\int_{\Omega}[\Phi_{0}(\mu_{1}u_{1}+\mu_{2}u_{2})]^{2}dx)^{1/2}$

Relation (24) and $\Phi_{0}\leq 1$ imply that

$\int_{\Omega}(\mu_{1}u_{1}+\mu_{2}u_{2})^{2}\Phi_{0}dx\geq\left|\Omega\right|\widehat{p}^{2}(t)$
(33)

The inequality $(\ref{27b})$ leads to

$\int_{\Omega}\psi(u_{1}+u_{2})^{2}\Phi_{0}dx\geq\psi\overline{\mu}^{-2}\left|\Omega\right|\widehat{p}^{2}(t)$
(34)

We replace $(\ref{27a})$ and $(\ref{27e})$ in the inequality $(\ref{27})$, we
obtain

$\widehat{p}^{\prime}(t)+(\lambda_{0}\overline{d}+c\underline{\mu}^{-1})\widehat{p}(t)\geq(\psi\overline{\mu}^{-2}-\lambda_{0}\alpha\overline{\mu}^{-1})\widehat{p}^{2}(t)$
(35)

taking $\overline{\tau}=\lambda_{0}\overline{d}+c\underline{\mu}^{-1};\
\underline{\psi}=\psi\overline{\mu}^{-2}-\lambda_{0}\alpha\overline{\mu}^{-1}$,
this gives

$\widehat{p}^{\prime}(t)+\overline{\tau}\widehat{p}(t)\geq\underline{\psi}\widehat{p}^{2}(t)$

Integrate the above inequality to get

$\widehat{p}(t)\geq\dfrac{\exp(-\overline{\tau}t)}{\dfrac{1}{\widehat{p}_{0}}-\dfrac{\underline{\psi}}{\overline{\tau}}(1-\exp(-\overline{\tau}t))},0\leq
t\leq T_{0}$ (36)

With this function $\widehat{p}(t)$,
$(v_{1}^{(0)},v_{2}^{(0)})=(\widehat{p}_{1}(t)\Phi_{0},\widehat{p}_{2}(t)\Phi_{0})$
is a lower solution of $(\ref{1})$ for every $T<T_{0}$ where $T_{0}$ is given
by the right-hand side of $(\ref{T0}).$ Since for
$\widehat{p}_{0}>\overline{\tau}/\underline{\psi}$ the function
$\widehat{p}(t)$ grows unbounded as $t\rightarrow T_{0}$, there exists
$T^{\ast}\leq T_{0}$ such that $\widehat{p}(t)\rightarrow+\infty$ as
$t\rightarrow T^{\ast}$. By relation $(\ref{21})$, $(u_{1},u_{2})$ must be
unbounded in $\left[0,T^{\ast}\right]\times\overline{\Omega}$, which implies
that the unique positive solution $\left(u_{1},u_{2}\right)$ possess the
blowing-up property $(\ref{15})$.

###  3.2 Discussion on the blow-up parameters

Following the notations in (18), we obtain the two conditions on the
parameters causing blow-up of the solution

  * •

If $0<\mu_{1}<\mu_{2}$ we choose $\psi_{1}=\mu_{1}b_{1}$ then

$\mu_{1}b_{2}<\mu_{2}b_{2}\Leftrightarrow\frac{\mu_{1}c_{1}+\mu_{1}b_{2}}{2}<\frac{\mu_{1}c_{1}+\mu_{2}b_{2}}{2}$

$\frac{\mu_{1}c_{1}+\mu_{1}b_{2}}{2}<\psi_{2}<\psi_{1}$

we obtain the first condition

$c_{1}+b_{2}<2b_{1}$ (37)

  * •

If $\mu_{1}>\mu_{2}>0,$ we choose $\psi_{1}=\mu_{2}c_{2}$ then

$\frac{\mu_{1}c_{1}+\mu_{2}b_{2}}{2}>\frac{\mu_{2}c_{1}+\mu_{2}b_{2}}{2}$

$\psi_{1}>\frac{\mu_{1}c_{1}+\mu_{2}b_{2}}{2}>\frac{\mu_{2}c_{1}+\mu_{2}b_{2}}{2}$

$\mu_{2}c_{2}>\frac{\mu_{2}c_{1}+\mu_{2}b_{2}}{2}$

we obtain the second condition :

$2c_{2}>c_{1}+b_{2}$ (38)

##  4 Conclusions

We have obtained the following conditions for the golbal existence of the
solution to the SKT problem :

let $2\alpha_{1}\lambda_{0}-b_{1}\leq 0$, $2\alpha_{2}\lambda_{0}-c_{2}\leq 0$
and
$(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}>0$
with $c_{2}b_{1}>c_{1}b_{2},$ If the following conditions hold

$\frac{-a_{1}(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})+a_{2}c_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq\frac{a_{1}c_{1}+a_{2}c_{2}}{c_{2}b_{1}-c_{1}b_{2}}$
$\frac{-a_{2}(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})+a_{1}b_{1}}{(2\alpha_{2}\lambda_{0}-c_{2})(2\alpha_{1}\lambda_{0}-b_{1})-b_{2}c_{1}}\leq\frac{a_{1}b_{2}+a_{2}b_{1}}{c_{2}b_{1}-c_{1}b_{2}}$
(39)

then the problem $(\ref{1})$ admits a unique global solution $(u_{1},u_{2})$
in $\overline{\Omega}\times[0,+\infty).$

For the blow-up conditions parameters, we have obtained :

if $\psi_{1}=\mu_{1}b_{1},\
\psi_{2}=\dfrac{\left(\mu_{1}c_{1}+\mu_{2}b_{2}\right)}{2}$ with
$\psi_{2}<\psi_{1}$ and $0<\mu_{1}<\mu_{2}$ then $c_{1}+b_{2}<2b_{1}.$

if $\psi_{1}=\mu_{2}c_{2},\
\psi_{2}=\dfrac{\left(\mu_{1}c_{1}+\mu_{2}b_{2}\right)}{2}$ with
$\psi_{2}<\psi_{1}$ and $\mu_{1}$ $>\mu_{2}>0$ then $c_{1}+b_{2}<2c_{2}.$

Under these conditions the solution of problem $(\ref{1})$ blows up as
$t\rightarrow T^{\ast}.$

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# A Passively Bendable, Compliant Tactile Palm with RObotic Modular
Endoskeleton Optical (ROMEO) Fingers

Sandra Q. Liu and Edward H. Adelson  Massachusetts Institute of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Many robotic hands currently rely on extremely dexterous robotic fingers and a
thumb joint to envelop themselves around an object. Few hands focus on the
palm even though human hands greatly benefit from their central fold and soft
surface. As such, we develop a novel structurally compliant soft palm, which
enables more surface area contact for the objects that are pressed into it.
Moreover, this design, along with the development of a new low-cost, flexible
illumination system, is able to incorporate a high-resolution tactile sensing
system inspired by the GelSight sensors. Concurrently, we design RObotic
Modular Endoskeleton Optical (ROMEO) fingers, which are underactuated two-
segment soft fingers that are able to house the new illumination system, and
we integrate them into these various palm configurations. The resulting
robotic hand is slightly bigger than a baseball and represents one of the
first soft robotic hands with actuated fingers and a passively compliant palm,
all of which have high-resolution tactile sensing. This design also
potentially helps researchers discover and explore more soft-rigid tactile
robotic hand designs with greater capabilities in the future.

##  I Introduction

When humans power grasp an object, they envelop their whole hand around the
object. Not only do the fingers curve around the object surface, but the palm
does also. In particular, the palm has a 1-DOF (degree of freedom) active
joint at the base of the four, non-thumb metacarpal bones. This joint, along
with the compliance of human skin, allows human hands to fully wrap itself
around various sized objects. Moreover, from a sensing perspective, the palm
can provide sensory information that the fingers or fingertips alone cannot
provide.

However, many existing robotic hands focus on the development of fingers, and
the palm is an afterthought. Very dexterous fingers or ones with a very large
bending radius can already slightly emulate this “wrapping” behavior of which
the palm is capable [1]. Other hand designs experiment with the thumb
abduction in the palm to replicate an enveloping grasp [2, 3]. Still, others
utilize a compliant and soft material as a palm to help passively enfold the
object being grasped instead of focusing on the palmar abduction joint [4];
the softness also serves as a safe way to interact with the world and other
people [5].

Figure 1: A novel dual compliant palm with three novel ROMEO fingers in a 120∘
configuration grasping an orange. Both the palm and the fingers have GelSight-
inspired tactile sensing through usage of the newly designed flexible
illumination system. To the right of the image are the resulting difference
images of the bumpy orange skin sensed by the palm and the orange stem sensed
by the ROMEO finger tip.

In general, we want a palm-like structure that is able to comply to and encase
the objects that are being grasped. We can do this using a soft material atop
a passive, compliant structure, which emulates a 1-DOF joint. This compliant
palm, along with tactile, soft-rigid fingers, will be able to help the
resulting hand perform a better enveloping grasp than if the palm were not
compliant. Moreover, palms can benefit from tactile sensing, which will
provide more information for object or contact classification. Finally, to
enable ease of use and manufacturing, we want to simplify the hardware
components of the cost-effective camera-based tactile sensing we use and also
modularize the fingers.

We present the following contributions:

  * •

The development of a new low-cost, effective illumination system for camera-
based tactile sensors that can be easily integrated into different soft
bodies.

  * •

A RObotic Modular Endoskeleton Optical (ROMEO) finger design, which uses the
new illumination strategy.

  * •

The novel designs of passively bendable and compliant palms with high-
resolution tactile sensing that are capable of increasing the surface area
contact of a grasped object, which can be incorporated into a fully high-
resolution tactile sensing hand (Fig. 1).

##  II Related Work

###  II-A Robotic Hands and Palms

Most existing robotic hands focus on the development of the fingers, from
rigid to soft to a hybrid combination of both. In papers where a palm is
developed, most researchers look at the thumb abduction to help create a
cupping motion for grasping. For example, Park and Kim develop an open-source
anthropomorphic rigid robotic hand with a simple 1-DOF thumb abduction joint
[2]. Tangentially, Xiong et al. design a more complicated rigid thumb joint
with a helical gear set to allow it to perform better grasps on different
objects [6]. These rigid robotic hand designs, with the addition of the thumb
joint in the palm, allow the hands to grasp many different types of objects.
Similarly, many soft hands approach palmar actuation by solely focusing on the
thumb abduction. Both Fras and Althoefer, and Zhou et al. design a thumb
abduction joint for the palm using soft rubber materials, which can help
conform better to objects than their rigid material counterparts [7, 8]. For
the novel soft-rigid hybrid gripper developed by Tavakoli and de Almeida [9],
the only palmar bending comes from the thumb joint bending towards the
direction of the other fingers. For the most part, a common theme of many hand
design papers are the lack of this 1-DOF actuation in the center of the palm.

Nonetheless, there are a few papers that explore palm concavity and changeable
palms. Pagoli et al. develop a linearly actuated palm that can extend to
engage with an object using suction or vacuum jamming [10], while Sun et al.
design a changeable palm which allows the fingers to rotate relative to the
central palm so that the fingers can more easily optimize a grasping pose [4].
From a more human-inspired direction, Wang et al. create a fully pneumatic
soft hand with a palm that is able to perform both palmar bending and thumb
abduction [11]. In a somewhat similar vein, Li et al. fabricate a passive
rotational palm joint with a jamming material palm that can engage when the
fingers press an object into the palm [12]. Capsi-Morales et al. also design
and manufacture a novel, rigid human-inspired hand that is capable of curling
the palm length-wise and can simulate a cupping grasp of the hand [13]. This
hand cupping motion is also replicated in a pneumatic soft hand developed by
Zhou et al., which has 26 actuated degrees of freedom [3].

All of these papers show that the inclusion of compliance, curvature, or a
changeable element in the palm helps to achieve better and more stable grasps.
However, they all require additional actuation, which can potentially make
them difficult to incorporate in other hand designs or to implement for
different hand configurations. They could also all be improved with the
addition of tactile sensing so that they can perform more identification or
contact classification tasks.

###  II-B Tactile Sensing for Robotic Hands

Although robotic hands have achieved a great variety of tasks using only
external vision [14, 15], there are situations where such cameras will not be
as useful, such as in a highly-occluded environment or digging through
clutter. In these scenarios, having tactile sensing is necessary.

A wide variety of tactile sensors for soft fingers and hands exists, ranging
from piezoresistive to capacitive to optical wave guides to pressure and
strain sensors [16, 17, 18, 19, 20]. Lei et al. also fabricate a biomimetic
tactile palm which is composed of an array of sensing electrodes and a
conductive liquid inside of a compliant skin [21]. However, many of these
sensors are low-resolution and potentially do not have the capability to
distinguish or classify an object with one touch and instead require multiple
touches [22].

One method for achieving high-resolution tactile sensing is to use camera-
based sensors, like the GelSight-inspired ones, which are simple, low-cost,
and effective [23, 24]. For these sensors, a camera looks at an illuminated
piece of elastomer atop a rigid structure, and the camera can track the touch
information on the illuminated surface. The simple concept of these sensors
allows for them to exist in multiple form factors, such as wedge shapes, round
finger-like domes, and curved claw-like structures [25, 26, 27, 28]. Even
though most of these sensors require a rigid backing to work, recent
developments have been made to integrate camera-based tactile sensors into
soft, compliant robots [29, 30, 31]. In particular, there was the development
of a soft-rigid finger with continuous high-resolution tactile sensing along
its front and sides [1]. Three of these fingers were placed into a gripper and
could classify objects using a singular grasp due to the rich tactile
information provided. However, the addition of a palm would perhaps greatly
improve its classification results and grasp stability, which it struggled
with for certain objects. To this end, we design this dual-compliance high-
resolution tactile sensing palm, which is then integrated into a fully tactile
hand.

##  III Methods

###  III-A Hardware

We develop a new camera-based tactile sensing illumination system and a novel
tactile, compliant palm, which can be used in a variety of different
configurations with our ROMEO fingers. These designs are shown in Fig. 2.

Figure 2: Rendered CAD files of the ROMEO finger (a) and the compliant palm
(b, c), showing the various labeled parts.

Illumination

Most GelSight-inspired sensors consist of a camera viewing an elastomer
illuminated by directional red, green, and blue (RGB) LEDs. This concept takes
advantage of the RGB color channels in a single camera image, meaning that a
single image of a tactile indentation allows the sensor to perform 3D
reconstruction using the color gradients. We wish to emulate this illumination
method so that it can more easily be applied in soft robots.

One of the issues with electronic integration into silicone soft robotic
bodies is the delamination that occurs when the silicone deforms around their
encased rigid electronics. This delamination creates an air interface between
the LEDs and the silicone, which can cause unpredictable light hot spots and a
lack of blended light gradients on the surface of the gel elastomer.

To mitigate this issue and to simplify the electronics, we turn to commercial
flexible 1.7 mm-diameter LED filaments (Adafruit), which are easy to use and
enclosed in silicone. The latter makes these strands easily to integrate into
silicone and the former enables them to flex with different soft bodies. Their
only downside is that the LED colors in one strand are not customizable,
making them difficult to use for different lighting configurations.

We turn to inspiration from the synthesized fluorescent silicone paint
formulated for the Baby GelSight Fin Ray [31], with the idea that if a section
of fluorescent paint covers a section of the blue LED strand, that section
will fluoresce the color we choose. Similarly, we can apply this concept to a
white light LED strand and use semi-transparent silicone paints. Furthermore,
because silicone bonds to itself, the paint will stay bonded to the LEDs, and
it will also stay bonded to the silicone gel body, preventing delamination.

The fluorescent paint, which follows the formula from [31] and uses acrylic
paint (Liquitex) and silicone adhesive, is applied to the visible sides of the
flexible LED filament after it is adhered to a surface. To get good
coloration, we use a Q-tip to apply at least a 1 mm surface of paint to all
visible sides so that the paint covers the blue LEDs.

On the other hand, for the white LEDs, we want to ensure that the paint is
semi-transparent so that the light can shine through the paint. As a result,
we utilize Silc Pig pigments (Smooth-On Inc), which are more translucent. Our
formulation is 10 parts of silicone adhesive mixed with 1 part of the red,
green, or blue silicone pigments, and then diluted with 3 parts NOVOCS Gloss
(Smooth-On Inc). We apply at most a 1 mm layer of paint.

This technique allows us to more easily use flexible LED filaments for
different configurations without the need to design a new PCB for every part
or component of our soft robot bodies. The materials also bond easily to one
another and can prevent delamination. We use these painted flexible LEDs in
both the ROMEO finger and compliant, tactile palm designs.

ROMEO Finger Design

We utilize a soft-rigid structure inspired by the bone and flesh structure of
a human hand. Doing so allows us to leverage the strength provided by the
structural rigidity and also allows us to take advantage of the soft
compliance of the gel. The inner structure is dubbed an “endoskeleton” and is
3D printed out of nylon and chopped carbon fiber material (Onyx, Markforged).
Using Onyx allows us to have a compliant hinge, which can easily be actuated
using tendons, and its carbon fiber content gives the endoskeleton structure
rigidity. Additionally, the endoskeleton is able to house both the camera and
LED structure, similar to how it was for [1].

Once the structure is 3D printed, Onyx braided line (Piscifun) is threaded
through 3D printed holes at the top surface of the endoskeleton. Afterwards, a
flexible LED filament is placed on top of the front and adhered using a 3D
printed PLA guide and a silicone adhesive. Once the silicone adhesive is
cured, the guide is removed from the structure.

Depending on which colored LED was incorporated, we mix the corresponding
silicone paints and apply them to the appropriate sections of the flexible LED
filament. In particular, each section of the filament surrounding the camera
holder has to exhibit three different colors: red, green, and blue.

After the LED filament is painted, the surface of the finger is coated in
silicone adhesive to help prevent delamination of the silicone from the
endoskeleton. Next, we prepare the mold for the silicone bodies that encase
the finger. Although we do not polish the mold, we make sure that the parts of
the mold for the camera mounts are covered with a single laser cut piece of
thin acrylic, so that the viewing ports are clear. A paint formulation of 1
part catalyst to 10 parts gray silicone pigment (Raw Materials Inc.) to 1.25
parts 4 $\mu$m aluminum cornflakes (Schlenk) to 30 parts NOVOCS Gloss (Smooth-
On Inc) is used for the tactile skin of the finger. This mixture is then
blended using an ultrasonicator to ensure even distribution of aluminum flakes
and gray pigment. The paint mixture is then airbrushed onto the top halves of
the 3D printed PLA molds, which have been treated with Inhibit-X (Smooth-On
Inc) and Ease Release 200 (Smooth-On Inc) for easy removal of the silicone.

We use XP-565 silicone (Silicones Inc.) because it is transparent, allowing
our camera to see through the entire gel structure. The mixture we use is 1
part silicone catalyst to 15 parts silicone to 3 parts plasticizer (Phenyl
Trimethicone, Lotioncrafter), which results in a softer gel. Before pouring
the silicone mixture into the prepared molds, we spread silicone adhesive in
between the various parts and screw the molds together to ensure that the
mixture will not leak. Once the mixture is poured inside, we leave the molds
on a vibrating plate for at least an hour to allow trapped air bubbles to
escape. These molds are then either left to air cure for 24 hours or they are
placed into a dehydrator at 50∘C for 6 hours. The lower cure temperature is to
ensure that none of the plastic elements reach their glass temperature and
cause the mold to warp. The main parts of the manufacturing process are shown
in Fig. 3.

Figure 3: The manufacturing process of both the finger and the palm, which
involve 3D printing the endoskeleton, prepping the flexible LED filaments,
painting and pouring the silicone, and finally demolding the parts. Not
pictured is the step where the aluminum paint mixture is airbrushed onto the
mold before silicone is poured.

The camera housings are designed such that when the finger bends at a 45∘
angle relative to itself, the two 120 field of view (FOV) cameras in each
finger segment are able to see a continuous tactile display of the entire
front and sides of the finger. We chose not to envelop the entire endoskeleton
structure to allow for a larger range of bending in the finger. The silicone
gel we use, although compliant, is unable to elastically compress on itself
even if grooves are added in between the finger links. By removing the gel in
between, we enable the finger to bend until the edges of the separate silicone
pieces jam against one another. Moreover, the silicone skin will not wrinkle,
as it would have if the endoskeleton was fully encased in silicone.

Compliant, Tactile Palm Design

For the palm design, we chose to incorporate two different types of
compliance: structural and material. This design choice was inspired by the
human palm, which exhibits both palmar bending at the base of the metacarpals
and has soft, compliant flesh on the outside. Having both compliance types
allows us to easily envelop various objects with the hand.

We considered having active actuation for the palm, but decided the
electronics would become too bulky, especially if each attached finger also
required separate actuation. To simplify electronics and keep the form factor
of the palm small, we decided to incorporate a type of passive compliance. One
solution that is easy to implement in different palm shapes is the cantilever,
which will deform if force is applied along its length.

In particular, if we place multiple cantilever beams with their ends all
concentrated in one area of the palm, we can achieve palmar bending passively
as long as an object is pressed into the palm. As a result, we choose to use
two cantilevers for ease of camera-based tactile sensing integration, but
acknowledge the fact that multiple cantilevers (n $\geq$ 2) can be used for
symmetry.

This passive structural compliance, along with the compliance from the gel
material enveloping the cantilever, will theoretically work better at
enveloping an object than just implementing structural or material compliance
by themselves (Fig. 4). For instance, if we take a cylindrical object and
press it against a rigid surface, we expect there to be only one line of
contact. However, if we integrate two cantilevers in the palm and press the
same object, we will get at least two lines of contact, one per cantilever
beam. The compliant gel will also provide multiple points of contact as it
elastically compresses and conforms to the surface area of the object. Thus,
when the two types of compliance are combined, we get the maximum surface area
contact, which is better for a more stable grasp. Not only do we have the
material gel compliance, we also get added surface area contacts when the
cantilever beams deform and help to better wrap the gel surface around the
area of the object.

Figure 4: The expected points of contact of a circle and rhombus object into
the different types of palms, ranging from no compliance to both structural
and material compliance. We expect having both types of compliance to give us
maximum surface area contact.

To design such a cantilever beam so that it does not plastically deform, we
assume a constant rectangular cross section as our baseline and use that to
implement a conservative hard stop at the base of the palm. We let $F$ stand
for force applied on the end of a cantilever of length $L$ and thickness $t$
with Young’s modulus $E$ and moment of inertia $I$. Then, we use the following
equations to determine the maximum deflection $\delta_{max}$ and stress
$\sigma_{max}$:

$\delta_{max}=FL^{3}/{3EI}$ (1)

$\sigma_{max}=FLt/2I$ (2)

Next, to find the conservative maximum deflection the beam can exhibit, we
combine (1) and (2) to get:

$\delta_{max}=\frac{2}{3}\sigma_{max}L^{2}/Et$ (3)

We use the Onyx material properties as an estimate for $\sigma_{max}$ and $E$
values, making sure that we set $\sigma_{max}$ as half of the max tensile
stress for the yield stress of the material and to account for a factor of
safety. Our $t$ value is determined by the thinnest, structurally sound 3D
printed beam we can obtain using the Markforged printer while $L$ is
determined by the reach of the ROMEO fingers in their respective
configurations.

Once the $\delta_{max}$ of the beam has been calculated, the resulting
geometry is designed and 3D printed using Onyx material, which can serve as a
flexible hinge. Because the maximum deformation of the beam keeps the material
in its elastic realm, we do not have to worry about plastic deformation over
repeated, multiple use.

After the palm base is printed, the flexible LED filament is adhered and
painted. Most of the steps to creating the gel pad are similar to the steps
for creating the ROMEO finger gels. The main difference is that the mold is a
single part instead of three separate parts. We also add two 3D printed parts
with an acrylic covering over the surface to the mold to ensure that there are
camera holes in the palm. The camera is placed into the cantilever afterwards
so that we can track the tactile deformation in the image without needing to
separate out the palm structural deformation.

The thickness of the gel pad for the palm is partially determined using the
camera FOV, so that the camera can see a relatively large portion of the palm
center. One camera is placed in each cantilever, and they are able to view
continuously along the central region of the palm.

Hand Structure

Each ROMEO finger has a base, which allows it to slot into different modular
palm configurations. To showcase this modularity, we also design multiple palm
configurations, all with both structural and material compliance (Fig. 5). We
also note that although we do not show it in this paper, the designed hands
can have fewer or more fingers.

Figure 5: Our ROMEO fingers and three different passively double compliant
palm configurations. From left to right, we have an anthropomorphic, a 120∘,
and a “Y” configuration gripper.

After we slot the fingers into the palm, the palm is then attached to the
electronic components and their housing, all of which are shown in Fig. 6.
Individual parts are connected to one another using M2 heated inserts.

Figure 6: The exploded view of our entire hand structure, including the
modular motor mounts and some electronics. Not pictured are the heated
inserts, bolts, wires, cameras, tendons, and flexible LEDs.

The PCB connects each of the flexible LED filaments in parallel with a 3 V
source. An adapter is custom designed for the palm so that it can fit atop a
modular motor mount, which houses a Dynamixel AX-12A motor. We specifically
design a modular motor mount so that it can be used for different hand
configurations and for multiple fingers. The motor is attached to the base of
each mount, which has rotationally symmetric mounting holes, allowing the
mounts to be arranged in the proper orientation for arbitrary finger
configurations.

###  III-B Software

The software components mainly involve the tactile sensing. We use 120∘ FOV
Raspberry Pi spy cameras and connect each of them to a separate Raspberry Pi
board. The camera images are streamed using mjpg-streamer and processed with
OpenCV and the fast Poisson solver [32, 33].

To get 3D reconstruction of a single tactile image, we take a reference image,
an image devoid of any tactile imprints, and subtract it from the tactile
image. This difference image is then processed and we get a resulting
uncalibrated 3D reconstruction (Fig. 7). We can also use this method for a
two-colored light, such as the lighting used in half of the palm design,
although the reconstruction will not be as accurate without the addition of a
neural network [25].

Figure 7: A comparison of the two different illumination systems with an M6
screw and a LEGO piece pressed into the gel surface. Both illumination
strategies result in decent 3D reconstruction images. For this paper, we chose
to use blue LEDs instead of the warm white LEDs.

##  IV Experiments and Results

###  IV-A Tactile Sensing

To test and compare the acuity of the two different tri-color lighting
systems, we integrate two ROMEO fingers, one with the blue LED and one with
the white LED. We capture videos of the tactile surfaces with objects pressed
into them and perform 3D reconstruction on individual objects.

We found that both the blue LED and white LED illumination methods seem to
yield similar 3D reconstruction results (Fig. 7). Both lighting strategies
were able to capture details in the individual threads of an M6 screw thread
and the words on a LEGO piece connector. However, the white LED had a less
even distribution of the RGB colors. As a result, we chose to use blue LEDs
and incorporate them into the ROMEO fingers and the passively compliant palm.

###  IV-B Paint Test

We observe the surface area coverage of a grasped object by implementing a
paint test [13]. The objects we use to test the palm are plastic shape sorting
toys with constant profiles (Fisher-Price), and we slather their surfaces with
paint. Before the paint dries, we press the side of the object profile into
the palm. In total, we use four different shapes: a cylinder, a cube, a plus
sign, and a star. We chose these objects for their simple form factors and
acknowledge that in the future, varying objects of different shapes and sizes
should be used for further rigor. Further, we note that adding fingers to
perform direct grasps could vary results based on finger configurations or
lengths, so we performed our experiment solely to test the palm by itself.

The different palms we test on are a non-compliant palm, a compliant structure
palm, a compliant gel palm, and a palm with both structural and material
compliance. All of the palm structures are printed on the Markforged with Onyx
material. For the palms with compliant gels, we use a one part mold.

Each object is pressed into the palms on a convex corner, if such a corner
exists. The same contact point is used on all four palms and the resulting
paint presses are shown in Fig. 8.

Figure 8: The results of the paint test, where each of the objects were
slathered with paint and then pressed into the four different palms. We can
visually verify that having both a compliant structure and the compliant gel
yields the largest contact area. This result is further corroborated in Table
LABEL:tab:bleh.

Visually, we can verify that having both structural and material compliance is
significantly better than having just one form of compliance or no compliance.
Furthermore, we compare the pixel area that each paint imprint covers and
verify that having these two compliant systems gives us the most contact area
against the palm (Table LABEL:tab:bleh).

TABLE I: Paint Test Comparisons (Pixel Area) Palm Type | Cylinder | Cube | Plus | Star
---|---|---|---|---
No Compliance | 11380 | 5794 | 12956 | 12920
Compliant Structure | 42347 | 9415 | 24812 | 12281
Compliant Gel | 44247 | 33963 | 53420 | 33388
Both Compliant | 65135 | 45797 | 69142 | 51228

Having a larger area of surface contact means that the grasped object is in a
stabler grasp. Thus, our preliminary findings indicate that using both methods
of compliance improves the palm functionality.

##  V Conclusion

In this paper, we develop a new low-cost illumination system that can be
utilized for various soft robotic bodies. We then use this system to design a
camera-based tactile finger with an endoskeleton and novel high-resolution
tactile palm which incorporates two types of compliance: structural and
material. Our design allows the palm to be used in countless different
configurations, enabling us to explore a larger space of soft-rigid robotic
hand designs with tactile sensors.

Future work involves designing other types of fingers, which could allow the
hand to grasp larger objects, performing more rigorous testing of the palm,
and experimenting with the tactile sensing. We also want to be able to
streamline and test different hand configurations to optimize for different
manipulation tasks.

Overall, we find that implementing both types of compliance in the palm
allowed there to be a greater contact area for the grasped object against the
palm, creating a more stable grasp. Furthermore, the addition of a palm with
tactile sensing capabilities can greatly improve different in-hand tasks that
might need tactile sensing such as classification, object reorientation, or
the determination of contact geometries.

##  VI Acknowledgements

Toyota Research Institute, Amazon Science Hub, and the SINTEF BIFROST
(RCN313870) project provided funds to support this work. We would also like to
express extreme gratitude to Megha H. Tippur for her help in PCB design and
soldering. Last but not least, we would like to thank Yuxiang Ma, Laurence
Willemet, Leonardo Zamora Yañez, and Jerry Zhang for their helpful tips and
insightful conversations.

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