A Determining Form for a Nonlocal System

Lemma 2.1 ([13]).

The eigenvalues of the spectral problem

where e⁢(⋅)∈L2⁢(-1,1), are

Moreover,

Lemma 2.2.

For α∈[12,2), system (1.1) has a weak solution: for any T>0 given u0∈L2⁢(-1,1), there exists a solution u with

Equation (1.1) holds as an equality in L4/3⁢(0,T;H-α/2⁢(-1,1)), this means that for any v∈L4⁢(0,T;H0α/2⁢(-1,1)), we have

for almost every t∈[0,T]. Moreover, system (1.1) has a global attractor A in L2⁢(-1,1), obviously, there is a constant ρ1>0, such that

Theorem 3.1.

Let ρ=4⁢R, where R:=(h+1)⁢ρ1. Let v∈BXρ⁢(0)={v∈X:∥v∥X<ρ}. There exists a Lipschitz mapping W:BXρ⁢(0)→C⁢(R,L2) for every v∈BXρ⁢(0). If u* is a steady state of the nonlocal system (1.1), then the determining form is

with the following properties:

1. The vector field in the determining form (3.2) is a Lipschitz map from the ball ℬXρ⁢(0)={v∈X:∥v∥X<ρ} into X. Hence, equation (3.2) has a short time existence and uniqueness for initial data in ℬXρ⁢(0). Moreover, equation (3.2) has global existence and uniqueness for all initial data in the forward invariant set ℬX3⁢R⁢(Jh⁢u*)={v∈X:∥v-Jh⁢(u*)∥X<3⁢R}.

2. Every solution of ( 3.2 ), with initial data in ℬ X 3 ⁢ R ⁢ ( J h ⁢ u * ) , converges to the set of steady states of ( 3.2 ).

3. All the steady states of the determining form ( 3.2 ) that are contained in the ball ℬ X ρ ⁢ ( 0 ) are given by the form v ⁢ ( s ) = J h ⁢ u ⁢ ( s ) for all s ∈ ℝ , where u ⁢ ( s ) is a trajectory that lies on the global attractor 𝒜 of the nonlocal system ( 1.1 ).

Proof.

For the nonlinear term f⁢(u)=u-u3, there are positive constants k,β such that

To prove (i), we need the following claim:

By Claim 1, we have a Lipschitz mapping

There exists a w such that W⁢(v)⁢(t)=w⁢(t) for all t∈ℝ, where w⁢(t) is the unique solution of (3.5).

Next, we prove that equation (3.2) has a local solution through verifying its nonlinear term is Lipschitz.

Let

For v1,v2∈ℬXρ⁢(0) we have

Thanks to (3.1) and (3.12) we have

and

Then

and

Similarly, we have

For the first term on the right-hand side of inequality (3.6), with inequalities (3.7) and (3.8), we have

and

With (3.9), we have

Then the estimation of first term on the right-hand side of inequality (3.6) is

For the estimation of the second term on the right-hand side of inequality (3.6), we have

Then we obtain

So F⁢(v)=-∥v-Jh⁢W⁢(v)∥X2⁢(v-Jh⁢u*) is Lipschitz, and equation (3.2) has a local solution. Moreover, by (3.9) we can get

The dissipative property of equation (3.2) implies that

is forward invariant for all t≥0, which proves that equation (3.2) has a global solution, that is, (i) holds.

Assertion (ii) follows directly from the forward invariance of (3.10) and the dissipative property of (3.2).

To prove (iii), we note that when either v=Jh⁢u*, or v∈ℬXρ⁢(0) such that ∥v-Jh⁢W⁢(v)∥X=0, the right-hand side of equation (3.2) is zero. In the first case, since u* is a steady state of the nonlocal system (1.1), we have u*∈𝒜.

In the second case, to understand better, we need the following claim which states a connection between equation (1.1) and the determining form.

Now we have v⁢(t)=Jh⁢W⁢(v)⁢(t), i.e., v⁢(t)=Jh⁢(w⁢(t)), for all t∈ℝ, where w⁢(t) is a solution of (3.5). In this case, the solution of equation (3.5), w⁢(t), is a bounded solution of system (1.1). Therefore, from (2.2) we know that w⁢(⋅) is a trajectory on the global attractor 𝒜 of system (1.1). Conversely, since ρ=4⁢R, it follows from (3.9) that J⁢(𝒜)⊂ℬX3⁢R⁢(Jh⁢u*)⊂ℬXρ⁢(0). Thus, for every trajectory u⁢(⋅)⊂𝒜 it follows from Claim 2 and (3.5) that u⁢(t)=W⁢(Jh⁢u)⁢(t) for all t∈ℝ. In particular, Jh⁢u=Jh⁢W⁢(Jh⁢u). Consequently, Jh⁢u is a steady state of (3.2) in ℬXρ⁢(0). Thus (iii) holds. ∎

Proof of Claim 1.

We divide the proof into two steps.

Step 1: A priori estimate. We take the inner product of equation (3.5) with w to obtain

where we used inequalities (2.1), (3.1), (3.3), the Cauchy–Schwarz inequality and Young’s inequality. Dropping the term in |w|4, we get

By Gronwall’s inequality [17] and assuming that ∥w∥ is bounded, we obtain

It follows that

For ∥δ∥, let γ=v1-v2. Then we have

Taking the inner product with δ, we obtain

Note that by the bound on the derivative of f we have

Using inequalities (2.1), (3.1), the Cauchy–Schwarz inequality and Young’s inequality, we get

Then we have

So by Gronwall’s inequality, we get

Thus it follows that

Step 2: Existence of solution. Applying the Galerkin method (see e.g. [17]), we get the existence and uniqueness of the solution for equation (3.5). Then by the above a priori estimate, we know that equation (3.5) has a global solution. Thus Claim 1 holds. ∎

Proof of Claim 2.

Let δ=w-u. The difference of (3.11) and (1.1) is

Suppose there is a time t*∈ℝ such that δ⁢(t*)≠0. Since δ⁢(t) is continuous, we can find the maximum interval (t0,t1) containing t*, such that δ⁢(t)≠0 for all t∈(t0,t1) and δ⁢(t0)=0. Taking the inner product of (3.14) with δ and using inequality (2.1), we have, for all t∈(t0,t1),

Using inequalities (3.1) and (3.13), we get

i.e.

Using Gronwall’s inequality, we obtain

where t0<σ0<t<t1. Taking σ0→t0+, by (3.4) we have δ⁢(t)=0 for any t∈(t0,t1). In particular, δ⁢(t*)=0, a contradiction. ∎

Remark 3.2.

Suppose system (1.1) has a more general nonlinearity f⁢(u), which is a C1 function satisfying

and

Then, for the case that α∈[1-2p,2), we could prove that Theorem 3.1 still holds. Note that, in this case, system (1.1) has a weak solution and a global attractor 𝒜 in L2⁢(-1,1).