Global Existence of Solutions of the Gierer-Meinhardt System with Mixed Boundary Conditions ()
1. Introduction
Biological spatial pattern formation is one area in applied mathematics under- going vivid investigations in recent years. Most models involved in biological phenomena are of the general reaction-diffusion type considered by Turing [1] . The distinctive attribute of Turing’s approach was the role of autocatalysis in coexistence with lateral inhibition. These studies led to the assumption of the existence of two chemical substances known as the activator and the inhibitor [2] [3] .
One of the famous studied models in biological spatial pattern formation is the Gierer-Meinhardt system which has received numerous attention and has been extensively studied [4] [5] [6] . The Gierer-Meinhardt system was used to model the head formation of a small, fresh-water animal called hydra [4] . We consider an activator concentration A and an inhibitor concentration H, satisfying the activator-inhibitor system given by
(1)
where
,
and
is a bounded smooth domain;
is the Laplace or diffusion operator in
;
is the unit outer normal at
,
is the directional derivative in the direction of the vector
. We assume that the reaction exponents
satisfy
(2)
The diffusion constants are
and
for the activator and inhibitor respectively. The time relaxation constant
was mathematically intro- duced due to its usefulness on the stability of the system. The constant b provides additional support to the inhibitor and may be thought of as a measure of the effectiveness of the inhibitor in suppressing the production of the activator and that of its own. In [7] , the ratio in the middle of (2) is called net self-activation index, since it compares how strongly the activator activates the production of itself with how strongly it activates that of the inhibitor. On the other hand, they call the ratio on the right hand side of (2) net cross-inhibition index, since it compares how strongly the inhibitor suppresses the production of the activator with that of itself. For the the inequality in (2), we expect the production of the activator to be severely suppressed by the inhibitor.
In [4] , some biological applications such as modeling of skeletal limb development, Robin boundary conditions are more realistic since the Neumann boundary conditions. A comparative numerical study of a reaction-diffusion system was made in [8] with a range of different boundary conditions and it revealed that certain types of boundary conditions selected a particular pattern modes at the expense of others. It was shown that the robustness of certain patterns could be greatly enhanced and the authors showed a possible ap- plication to skeletal pattern of limb.
Special case was considered for the Neumann boundary condition (i.e.
) in [5] [9] . In [5] , Masuda and Takahashi proved the global solutions of the special case of (1) with b = 0 exists for
provided in addition to (2) one has
, we note the strict inequality here. In [9] , Jiang improved the net self-activation index noted in [5] to
and showed that the solutions exists globally in time.
In this paper we consider the Robin boundary condition (a ≠ 0) on the activator and Neumann boundary condition on the inhibitor and study the global (in time) existence of solutions for the Gierer-Meinhardt system in (1). The theorem and lemmas in this current manuscript are inspired by [9] . We establish the global (in time) existence of (1) by proving the theorem below:
Theorem 1. Suppose
is a smooth bounded domain with a smooth
boundary
in
. Assume that
. Let
and
,
. Then every solution
of (1) exists globally in time.
2. Proof of Theorem 1
The local existence and uniqueness of (1) is standard and more details can be found in [10] [11] . A priori-estimates need to be ascertain in order to prove global in time existence of solutions. Let
be a solution of (1) in
. We want to ascertain that H is bounded away from zero. Let
then
Lemma 1.
for all
.
Proof. Let
(3)
then
satisfies
and
but
thus
Additionally at
, from (3)
So
for any
.
Hence from maximum principle,
in
and thus
□
Lemma 2. For any two constants
, let
.
Define
Suppose
(4)
then
(5)
Here
(6)
and
(7)
where
,
,
and
Proof. Let
and
,
But
Additionally,
now we have,
We deduce from above a quadratic equation involving
and
. Let us
fix
and choose
, we have
therefore the quadratic form involving
and
in the inequality above
is non-positive since its determinant
Thus
We have
and
we choose
sufficiently small such that
and
Now, we write
but
thus
where
and
are defined by (6) and (7)
by Young’s inequality, we obtain
Therefore
but by Hölder’s inequality
Thus
Finally,
□
Remark 1. The condition in (4) is true for any
where
Lemma 3. Let
,
and
on
be an integrable function. Let
be a nonnegative function on
satisfying the differential inequality
(8)
Then
(9)
where
is the maximal root of the algebraic equation
Moreover, if
, we have
(10)
where
is the maximal root of the algebraic equation
Proof.
(11)
Let
and
in particular, at
we obtain now that
(12)
Notice that the quantity
is finite and hence (9) follows from (12). As
in (11), we ascertain
thus (10) follows since
is finite. □
The next Lemma follows after applying Lemma 3 to (5).
Lemma 4. For any
such that
, and all conditions in Lemma 2 hold true. Then there exists a constant
such that
(13)
for all
.
Proof. For sufficiently small
, such that
with
, we obtain
therefore we deduce from Lemma 2 that
satisfies
Since
and
for all
, then from Lemma 3, (13) is true for
and sufficiently small
such that
. Since H is bounded away from zero, then (13) is true for any
. □
From Lemma 1 and Lemma 4, we deduce the Corollary below.
Corollary 1. Let
and all other assumptions in Theorem 1, Lemma 2, Lemma 3, and Lemma 4 hold true. Define
then there exist positive constant
, such that
for all
.
Proof. The proof to this Corollary follows from Lemma 3 and Lemma 4. □
3. Conclusion
In this paper, we have studied the Gierer-Meinhardt system with Robin boundary conditions and Neumann boundary conditions on the activator and inhibitor respectively. Global existence of solutions have been obtained under the mixed boundary conditions using a priori estimates of solutions.