Existence and Uniqueness of Almost Periodic Solution for a Mathematical Model of Tumor Growth ()
1. Introduction
Since 1980, researchers in mathematics and biology have proposed and studied several deterministic mathematical models for tumor growth by diffusion equation. The main assumption is that these models serve as simplified but complementary description for one aspect of a complex biological phenomenon: the growth and stability of tissue. Given some simplified conditions, the study of these models is focused on describing qualitatively the early stages of growth of tissue. The diffusion equation is the type of a linear partial differential equation.
For example, Shymko, Glass [1] [2] and Adam [3] [4] proposed the following model governed by an inhomogeneous diffusion equation
(1)
with the source term
for
and 0 otherwise. Here L is the initial length of the chalone-producing tissue, being confined to the domain
. The production rate of the chalone is
per unit length, the diffusion coefficient is D and the decay rate is
which is proportional to its concentration
. Since
is not constant, the source of mitotic inhibitor is not uniformly distributed within the tissue (in contrast to many earlier results). It was found that stable and unstable regimes of growth become significantly modified from the uniform-source case. Consequently, this model is very sensitive to the type of source term assumed. In fact, many of the existing deterministic models of tumor cell growth are proposed by an ordinary differential equation (ODE) coupled to one or more equations of reaction and diffusion type. The ODE derives from mass conservation applied to the tumor and describes the evolution of the tumor boundary and the reaction-diffusion equations describe the distribution of nutrients (oxygen and glucose) and growth inhibitory factors (chalones) [5] [6].
Clearly, a more realistic model requires a higher dimension because systems governing tumor growth are best served in a three dimensional domain. Consequently, Britton and Chaplain studied a more generalized system below [7].
(2)
(3)
(4)
where
is the concentration of some chemical inhibitors in a bounded n-dimensional region
(
). We note that
, P is the permeability of the tissue surface. Using maximum principles for parabolic and elliptic operators, the authors examined the effect of growth inhibitory factor. It was shown that if
and
satisfy the conditions of the parabolic comparison theorem, then C is always non-negative and unique. Also, the concentration decreases monotonically in the open interval
provided that
and f differentiable with
. This model is certainly a big improvement compared to one-dimensional system.
The following model adds a more general, time-dependent source function
(5)
(6)
(7)
Here the source function is time-dependent. The boundary condition is of homogeneous Robin type. Existence and uniqueness of an almost periodic solution for this model were studied in [8] with the following result. If
and
,
is local Lipschitz continuous,
-norm of K is uniformly bounded in time, and S is temporally almost periodic with its
-norm uniformly bounded in time. Then there exists a unique almost periodic solution for (5)-(7).
In this paper, we study a similar system similar to (5)-(7) under non-homogeneous Robin boundary condition
in
and initial condition
. We prove that, under certain conditions of initial and boundary data, there exists a unique solution which is almost periodic. The existence is obtained via continuous contraction semigroup and fixed point theorem and uniqueness is obtained via integral estimates on L2 norm of C.
2. Existence and Uniqueness of the Almost Periodic Solution
In this section, we discuss the existence and uniqueness of almost periodic solution when the source function is almost periodic and the dynamics of the system. We consider the following system with a time-dependent source function
and Robin inhomogeneous boundary data:
(8)
(9)
(10)
Here
,
for any
. First, we prove the following existence theorem for (8)-(10).
Theorem 2.1. Assume that
and
. Let
be local Lipschitz continuous,
-norm of K is uniformly bounded in time, and S is temporally almost periodic with its
-norm uniformly bounded in time. Then there exists a unique almost periodic solution for (8)-(10) such that
for any
.
PROOF. We use a transformation
where H is a smooth function in
satisfying the boundary condition
on
. Then (8) is converted to
(11)
where G satisfies the homogeneous boundary condition
. Similar to the system studied in [8], local existence for (11) can be obtained by [9] [10] [11]. Consequently this establishes local existence for (8)-(10).
Let
be the
norm of S and
be the
norms of C and
. We differentiate
and substitute (10) to get
(12)
Since
-norm of K is uniformly bounded in time, there exists a number
such that
(13)
Substitute (13) in (12) we have
(14)
This implies that
(15)
when
for some sufficiently large T. Therefore, all solutions C enter the following bounded set in
(16)
Suppose that
and
are two solutions with same initial value
and boundary value
. Then
satisfies the following system
(17)
(18)
with
.
A quick calculation shows
(19)
This implies that
(20)
therefore we know
and the solution is unique since
.
On the other hand, if we assume that
and
are two solutions with
as initial values respectively and same boundary value. Then
satisfies (17) and (18) with initial value
. It is easy to check that C satisfies (19) and (20). Therefore,
(21)
Define the solution operator
by
for
, where
is the solution of (8)-(10). By (21),
possesses strong contraction property with absorbing sets (16).
Recall that a function
where
is a metric space, is called almost periodic [12] [13] if for every
there exists a relatively dense subset
of
such that
(22)
for all
and
. Almost periodic functions play an important role in the theory of nonautonomous dynamical systems.
By similar arguments in [14] [15] [16], Theorem 2.2 in [8] holds for system (8)-(10) and the corresponding pullback attractor defines a unique almost periodic solution. Therefore the proof of Theorem 2.1 is now completed.
3. Conclusion
We study a mathematical model of tumor growth presented by a diffusion equation with appropriate initial and boundary conditions. The boundary value is of Robin type and is inhomogeneous. We show that there exists a unique solution that is almost periodic. The main method is to show that a pullback attractor defines a unique almost periodic solution for the system.
Acknowledgements
This research was supported by William R. Kenan Jr. Professorship, Staley Small Grant and Wellesley College Faculty Award.