The Global Attractor and Its Dimension Estimation of Generalized Kolmogorov-Petrovlkii-Piskunov Equation ()
1. Introduction
Many scholars at home and abroad have studied the dynamical system theory described by mathematical physics equations, such as Navier-Stokes equation, nonlinear Schrödinger equation, KdV equations, reaction-diffusion equation, damped semilinear equation, etc, and estimated the dimension of the attractor.
In [1] , Xu et al. studied the type KPP equations of (3 + 1) and (1 + 1) dimensions:
Which is the exact solution of the equation.
Wu studied the initial boundary value problem of generalized KPP equations in [2] :
where
is the diffusion coefficient,
is the reaction rate constant and
is the time delay constant. When
, it became the famous KPP equation:
The traveling wave solution is studied and the propagation speed is
.
In [3] , Chen et al. proposed a new auxiliary equation method to find the exact traveling wave solution of the nonlinear development equation. By selecting Bernoulli equation with variable coefficient as the auxiliary ordinary differential equation, the generalized Burgers-KPP equation was solved according to the principle of homogeneous equilibrium, and the traveling wave solution of the equation was obtained. In [4] , Cao et al. studied the stability and uniqueness of the generalized traveling wave of discrete Fisher-KPP equations with general time and space dependence. More studies on global attractors and their dimension estimation of equations can be seen in references [5] - [10] .
In this paper, we consider the initial boundary value problem of the following generalized KPP equations:
, (1)
, (2)
(3)
where,
, p is a natural number.
Notation is introduced for the convenience of narration:
represents the norm in
space;
and
represents the norm and inner product in
space, and
.
2. The Existence of a Global Attractor
In order to prove the existence of problems (1)-(3) global attractors, the following conclusions are needed:
Lemma 1. Let
, then the solution u of problems (1)-(3) is estimated as follows:
,
, (4)
where
, C is a normal number that depends on
, and
.
Proof. By taking the inner product of both sides of Equation (1) with u, we get:
(5)
Obtained from Formula (5):
. (6)
Thus, there:
. (7)
By Young’s Inequality, there:
(Let
). (8)
By substituting Formula (8) into Formula (7), we get:
(Let
). (9)
Due to
, so:
. (10)
From Gronwall’s inequality, we get:
, (11)
(among them
).
Hence,
.
Lemma 2. Let
, then the solution u of problems (1)-(3) is estimated as follows:
,
, (12)
where
,
is a normal number that depends on
, and
.
Proof. By taking the inner product of both sides of Equation (1) with
, we get:
. (13)
Obtained from Formula (13):
,
Thus, there are:
. (14)
Due to
, (15)
So,
(because
). (16)
And because:
. (17)
Obtained from Formula (14):
. (18)
By (16)-(18), get:
. (19)
From the Gronwall inequality, obtain:
,
(Ream
). (20)
Therefore,
.
Theorem 1. Set a given function
, and
, then the problems (1)-(3) has a unique solution u, such that
.
Proof. 1) Existence: According to Lemma 1 and Lemma 2, the solution
of problems (1)-(3) exists.
2) Uniqueness: Let
be two solutions of Equation (1), and let
, then:
, (21)
. (22)
Obtained by (21) and (22):
, (23)
where
.
Take the inner product of both sides of Equation (23) with
, we get:
.
Thus, have:
(24)
And because:
, (25)
. (26)
Obtained by (24)-(26):
,
Thus, there are
. (27)
So, from (27), we get:
, that is
.
Define. [7] Let
be a continuous operator semigroup,
,
;
.
If the compact set
is satisfied:
1) Invariance: A is an invariant set under the action of a semigroup
, i.e.
.
2) Attraction: A attracts all bounded sets in
, that is, any bounded set
, have:
.
In particular, when
, all orbitals
from
converge to A, that is:
.
Then, the compact set A is called the global attractor of semigroup
.
Theorem 2. [8] Let E be a Banach space,
be a family of operators,
,
,
,where I is the identity operator. Let
satisfy:
1)
is bounded, that is,
, then there is a constant
such that
.
2) There is a bounded absorption set
, that is, any bounded absorption set
, and there is a constant
time such that the bounded absorption set
.
3) For
,
is a completely continuous operator.
Then, the semigroup
has a compact global attractor A.
Theorem 3. If the problems (1)-(3) have A solution and satisfies the conditions of Lemma 2, then the problems (1)-(3) have a global attractor A, that is, there is a compact set
such that:
1)
.
2) Any bounded set
, yes:
.
In particular, when
, all orbitals
from
converge to A, that is:
.
Proof. Let
, and
, then follows from Lemma 1:
.
Thus, there is:
. (28)
Ream
, then
.
Take
, then
, Formula (28) can be written as:
.
Similarly, from Lemma 2:
.
Thus, have:
. (29)
Ream
, then
.
Take
, then
, Formula (29) can be written as:
.
Let
, and u is bounded in
, and
is tightly embedded in
, so B is the compact absorption set in
.
Let
, and
be the corresponding two solutions of the equation,
, and let
, then w satisfies:
From uniqueness, can be obtained:
.
Thus, have:
.
So, the operator
is continuous.
Thus, from Theorem 2, we know that problems (1)-(3) exist global attractors:
.
3. The Dimension Estimation of the Global Attractor
In order to establish the Hausdoarff dimension of the problems (1)-(3) global attractor A, the upper bound of the fractal dimension. A linear variational problem for problems (1)-(3) needs to be established:
, (let
),
i.e.
, (30)
, (31)
where
is solution of problems (1)-(3) with
.
Lemma 3. Let
and
,
, then the linearization problems (30) and (31) have unique solutions:
. (32)
In addition, remember
, then
, there is a constant E related to R and T, such that:
, (33)
where
, this shows that the operator
is uniformly differentiable on A, and that the differential of
in
at
is:
.
Let
, and W is positive definite dense, then
exists and is bounded.
So, (30) becomes:
.
Multiply both sides by
, you get:
, (34)
where
.
Let’s say
is m solutions of (30) (31), and the corresponding initial values are:
, then:
Let
represent the orthogonal projection on the space spanning
to
.
Next, the exponential attenuation of the m dimensional volume element
and the dimension estimation of the global attractor A are considered. Thus, there is:
Theorem 4. Let the global attractor
of problems (1)-(3), and m satisfy:
, (among which
,
,
).
Then:
1) When
, the m dimensional volume element
vwill decay exponentially.
2)
.
Proof. Let’s say
is a set of orthogonal bases for
and satisfies:
,
where
is the eigenroot of the operator
.
The lower bound of
is estimated below:
(35)
Again
, so (35) can be written as:
. (36)
Because
, then there is
.
Therefore, Formula (36) becomes:
.
That is
, (among others
). (37)
Let
, then Formula (37) becomes:
. (38)
Thus, have:
. (39)
And
, and the series
converges, let’s say the series
converges to the normal number
, so Formula (39) becomes:
. (40)
When
, have
, i.e.
, (among them
), thereby having
.
Let
be the Lyapunov exponent, then there is an inequality:
,
So,
, and
.
Therefore,
.
4. Closing Remarks
In this paper, the existence and uniqueness of the solutions (1)-(3) of the initial boundary value problem of generalized KPP equation and the existence of the global attractor are studied. The Hausdorff dimension and fractal dimension of the global attractor are estimated.