反応拡散方程式系におけるパターン形成に対する計算機援用証明

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

反応拡散方程式系におけるパターン形成に対する計算機援用証明

蔡, ??

九州大学大学院数理学府

https://doi.org/10.15017/21706

出版情報：Kyushu University, 2011, 博士（数理学）, 課程博士バージョン：

権利関係：

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A computer-assisted proof for the pattern formation on reaction-diﬀusion systems

Shuting Cai

Graduate School of Mathematics, Kyushu University

2012

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Abstract

In this paper we give a method by computer-assistance to prove a pattern formation. As a typical model we consider two dimensional time-dependent reaction-diﬀusion equations with Neumann boundary conditions. For suitable system parameters we solve (approximately) the parabolic problem, hop- ing for some convergence to some pattern formation stationary (approximate) solution, and improve the approximation to a stationary solution by Newton’s method, then enclose the stationary solution by our numerical veriﬁcation method. Next we prove that the operator linearized at the exact stationary solution is a sectorial operator and compute a bound for the resolvent of the linearized operator which is needed for semigroup estimates. By using the semigroup estimate we analytically compute a domain of attraction for the stationary solution, i.e. some (norm-)neighborhood of the stationary solution such that, for initial data within this neighborhood, the parabolic solution converges to the stationary solution. For suitable initial conditions, if we enclose the solution of the parabolic problem until, for some time T, the enclosing set is a subset of the domain of attraction, then we can conclude that from time T on, convergence to the stationary solution takes place. This gives a complete convergence result, proving a pattern formation, for the initial conditions used for the parabolic problem.

Keyword:

Pattern formation; Computer-assisted proof; Reaction-diﬀusion system;

Domain of attraction.

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Chapter 1 Introduction

We consider a time-dependent reaction-diﬀusion system with Neumann bound-

ary conditions 





u_t=γf(u, v) + ∆u in Ω, v_t=γg(u, v) +d∆v in Ω,

∂u

∂ν = ∂v

∂ν = 0 on∂Ω,

(1.0.1)

where Ω is a bounded domain in ℜ², γ, dare some positive constants,f, g are nonlinear/linear functions depending on each model and ν = (ν₁, ν₂) is an outer unit normal vector on ∂Ω.

This kind of problem (1.0.1) can be applied in mathematical biology[10, 11]. One simple system is FitzHugh-Nagumo reaction[6, 15]. In [7], u as an activator and v as an inhibitor are interpreted as relative concentrations of two substances known as morphogens. In [26], this kind of problem, called

”Excitable media”, can be applied in mammalian heart muscle and its cells and Xenopus eggs. For more background, see references in [23].

There are many papers considering the stationary solution of the system.

Here we mention some results on the systems of FitzHugh-Nagumo type.

There are several results about Dirichlet problem[4, 3, 20, 24]. In [4] the minimization problem associated with the system is considered, and in [3]

the peak solutions for the system are investigated. There is also one paper considering the relationship between the parameter and the solution of the system[20]. In [24], by using a numerical veriﬁcation method, the author encloses the exact solution of the equations. There are also some results for Neumann problem. In [5] some results on the solutions with interior and boundary peaks are shown, and in [21] the relationship between one parameter and the energy minimizers is discussed. In [2], the authors proposed a numerical veriﬁcation method to enclose a solution of the two dimensional

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system.

In this paper, we will propose a computer-assisted method to prove a pattern formation. Knowing the pattern formation is very important in biology[11]. When we solve a time-dependent problem, we often see some numerical convergence of them, but analytically there is no proof of it. On the other hand when we compute a steady state problem, we do not know from which initial state the stationary solution appeared. This paper is about how to overcome these two diﬃculties. As a concrete example, we will apply our method to Schnakenberg equation, which is another reaction-diﬀusion system.

There are eight chapters in this paper. In chapter 2, we prepare some function spaces and notations and then the ﬁxed-point formulation and the construction of a priori error estimate for the projection are derived. And in Chapter 3, for suitable system parameters we solve (approximately) the parabolic problem and improve the approximation to a stationary (approximate) solution by Newton’s method. After that, in Chapter 4, we use Nakao’s method to enclose the stationary solution near this approximate solution.

This method, which is similar to the method in [2], is based on the infinite dimensional fixed point theorem. First, the time-independent system is rewritten in a fixed point form and then the fixed point equation is decom- posed into the finite dimensional part and the infinite dimensional error part.

Based on these two parts, we construct a set which satisfies the hypothesis of Schauder’s fixed point theorem for a compact map in a suitable Sobolev space. This verification method was originated by Nakao[16] and then has been developed by him and his coworkers[18, 19, 12, 13, 14].

Then we prove that the operator linearized at the exact stationary solution is a sectorial operator and compute a bound for the resolvent of the linearized operator which is needed for semigroup estimates. By using the semigroup estimate we analytically compute a domain of attraction for the stationary solution. And thus, for suitable initial conditions, if we enclose the solution of the parabolic problem until, for some time T, the enclosing set is a subset of the domain of attraction, then we can conclude that from timeT on, convergence to the stationary solution takes place. This is showed in Chapter 6. And in order to get the results in Chapter 6, we propose a computer-assisted method to exclude the eigenvalues of the linearized operator in Chapter 5. In Chapter 7, there are some numerical results. We apply our method to an example with some suitable system parameters and then get its domain of attraction. At last there are some conclusions in Chapter 8.

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Chapter 2 Some notations and projection error es- timation

We consider the domain Ω = (0, l)×(0, l) ⊂ ℜ²(l ≥ 1). Then we choose basis function as

φ_i₁_i₂(x, y) = cos(i₁πx/l) cos(i₂πy/l), i₁, i₂ = 0,1,2, . . . For a ﬁxed non-negative integer N we reorder the basis function

φi = cos(i1πx/l) cos(i2πy/l),(i= 1,2,3,· · ·(N + 1)²) (i1, i2 = 0,1,2,· · ·N) in the following way:

i=i₁(N + 1) +i₂+ 1.

The Sobolev spaceW^k,p(Ω)(1≤p≤+∞) is deﬁned as W^k,p(Ω) :={u∈L^p(Ω) :D^αu∈L^p(Ω),∀|α| ≤k},

where α is a multi-index. The natural number k is called the order of the Sobolev space W^k,p(Ω).

And we suppose that for the L²-Sobolev space of order k on Ω, H^k(Ω), ψ ∈H^k(Ω) (k ≥0) is expanded in the Fourier series as

ψ =

∑∞ i=1

a_iφ_i (a_i ∈ ℜ).

For a non-negative integerN, let XN denote a function space as X_N :=

{ v_N =

∑N n,m=0

c_nmφ_nm|c_nm∈ ℜ }

⊂H²(Ω)⊂H¹(Ω).

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Forz₁, z₂ ∈H¹(Ω), deﬁne the usual inner product as

⟨z₁, z₂⟩H¹(Ω):= (z₁, z₂)_L²_(Ω)+ (∇z₁,∇z₂)_L²_(Ω), where (·,·)_L²_(Ω) means the inner product on L²(Ω).

Forz ∈H^k(Ω)(1≤k < ∞, k̸= 2), deﬁne the usual norm as

∥z∥²_H^k_(Ω) := ∑

|α|≤k

∥D^αz∥²L²(Ω).

When k= 2, for z = ∑^∞

n,m=0

c_nmφ_nm ∈H²(Ω), we have

∥z∥²L²(Ω)+∥∇z∥²L²(Ω)+∥∆z∥²L²(Ω)

=∥z∥²_L²_(Ω)+∥∇z∥²_L²_(Ω)+

∑∞ n,m=0

c²_nm(∥(φ_nm)_xx∥²_L²_(Ω)+∥(φ_nm)_yy∥²_L²_(Ω) + 2((φ_nm)_xx,(φ_nm)_yy)_L²_(Ω))

=∥z∥²_L²_(Ω)+∥∇z∥²_L²_(Ω)+

∑∞ n,m=0

c²_nm((n⁴+m⁴)π⁴/l⁴ + 2n²m²π⁴/l⁴)∥φ_nm∥²_L²_(Ω)

=∥z∥²_L²_(Ω)+∥∇z∥²_L²_(Ω)+

∑∞ n,m=0

c²_nm(∥(φ_nm)_xx∥²_L²_(Ω)+∥(φ_nm)_yy∥²_L²_(Ω) + 2((φ_nm)_xy,(φ_nm)_xy)_L²_(Ω))

= ∑

|α|≤2

∥D^αz∥²_L²_(Ω),

where (φ_nm)_x is the derivative of φ_nm with respect to x, (φ_nm)_xy is the derivative of (φ_nm)_x with respect to y and (φ_nm)_xx, (φ_nm)_yy are the second derivative of φ_nm with respect to x and y respectively, therefore, we deﬁne the norm in H²(Ω) as

∥z∥²_H²_(Ω):=∥z∥²_L²_(Ω)+∥∇z∥²_L²_(Ω)+∥∆z∥²_L²_(Ω). And for z₁, z₂ ∈H²(Ω), deﬁne the inner product as

⟨z₁, z₂⟩H²(Ω) := (z₁, z₂)_L²_(Ω)+ (∇z₁,∇z₂)_L²_(Ω)+ (∆z₁,∆z₂)_L²_(Ω). For z ∈ H¹(Ω), let P_N : H¹(Ω) →X_N denote the H¹-projection deﬁned by the truncation operator:

P_N

( ∑_∞

n,m=0

c_nmφ_nm )

=

∑N n,m=0

c_nmφ_nm,

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which satisﬁes

⟨z−P_Nz, z_N⟩H¹(Ω) = 0, for all z_N ∈X_N. When z∈H²(Ω), we also have

⟨z−P_Nz, z_N⟩H²(Ω) = 0, for all z_N ∈X_N. Therefore, we also use P_N as a H²-projection.

Set P :H²(Ω)×H²(Ω)→X_N ×X_N as

P(z₁, z₂) := (P_Nz₁, P_Nz₂), z₁, z₂ ∈H²(Ω).

For Hilbert spaces X and Y, we deﬁne the inner product and the norm in X×Y as

(( x₁ y₁

) ,

( x₂ y₂

))

X×Y

:= (x₁, x₂)_X + (y₁, y₂)_Y

and

( x y

):=

√∥x∥²X +∥y∥²Y.

Deﬁning the operator L : H²(Ω) → L²(Ω) by Lψ := −△ψ +ψ, we get the following lemma.

Lemma 2.0.1 For all ϕ∈L²(Ω), the linear equation { Lψ =ϕ in Ω,

∂ψ

∂ν = 0 on ∂Ω (2.0.1)

has the unique solution ψ ∈H²(Ω).

Proof. Let ψ = ∑^∞

n,m=0

ψ_nmφ_nm (ψ_nm∈ ℜ, n, m= 0,1,2, . . .).

Forϕ= ∑^∞

n,m=0

ϕ_nmφ_nm ∈L²(Ω) (ϕ_nm ∈ ℜ, n, m= 0,1,2, . . .),let

ψ_nm = ϕ_nm

1 +n²π²/l²+m²π²/l², then Lψ =ϕ holds. By observing

∑∞ n,m=0

(1 +n²π²/l²+m²π²/l² +n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴)ψ_nm²

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=

∑∞ n,m=0

(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴)

·

( ϕ_nm

1 +n²π²/l²+m²π²/l² )2

<

∑∞ n,m=0

3(l²+n⁴π⁴/l²+m⁴π⁴/l²) ϕ²_nm

1 +n⁴π⁴/l⁴+m⁴π⁴/l⁴

=

∑∞ n,m=0

3ϕ²_nml² <∞, and

∥ψ∥²_H²_(Ω)

=

∑∞ n,m=0

(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴)ψ²_nm∥φ_nm∥²_L²_(Ω)

<

∑∞ n,m=0

l²(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴)ψ_nm² , ψ ∈H²(Ω) follows.

Noting that the solution ofLψ =ϕ is characterized as the solution of

∫

Ω

∇ψ· ∇v dx+

∫

Ω

ψv dx=

∫

Ω

ϕv dx, ∀v ∈H¹(Ω),

we have ∫

Ω

(−∆ψ+ψ)v dx+

∫

∂Ω

∂ψ

∂νv dx=

∫

Ω

ϕv dx,

and thus ∫

∂Ω

∂ψ

∂νv dx= 0.

Sincev is arbitrary inH¹(Ω), in case ψ ∈H²(Ω) we have ^∂ψ_∂ν = 0 in the trace sense.

If there exists another solution ˜ψ ̸=ψ, we get Lψ˜=ϕ,

which means the existence of a non-trivial solution of the equation Lφ= 0.

But noting that

(Lφ, φ)_L²_(Ω) =∥φ∥²_H¹_(Ω)

holds, Lφ = 0 derives ∥φ∥²_H¹_(Ω) = 0, i.e. φ = 0, which contradicts the assumption and the uniqueness of the solution of Lψ =ϕ follows. 2

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Remark 2.0.2 For ϕ ∈ L²(Ω), we denote the solution of (2.0.1) as L⁻¹ϕ.

Then it is clear that L⁻¹ϕ satisﬁes Neumann boundary conditions.

Lemma 2.0.3 For all ϕ∈L²(Ω), the linear equation





(∆²−∆ +I)ψ =ϕ in Ω,

∫

∂Ω

(∂ψ

∂νv−∆ψ∂v

∂ν +∂(∆ψ)

∂ν v )

dx= 0, for ∀v ∈H²(Ω) has the unique solution ψ ∈H⁴(Ω).

Proof. Let ψ = ∑^∞

n,m=0

ψ_nmφ_nm (ψ_nm∈ ℜ, n, m= 0,1,2, . . .).

Forϕ= ∑^∞

n,m=0

ϕ_nmφ_nm ∈L²(Ω) (ϕ_nm ∈ ℜ, n, m= 0,1,2, . . .),let

ψ_nm= ϕ_nm

1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴, then (∆²−∆ +I)ψ =ϕ holds. By observing

∑∞ n,m=0

(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+n⁶π⁶/l⁶+m⁶π⁶/l⁶ +n⁸π⁸/l⁸+m⁸π⁸/l⁸+ 2m²n²π⁴/l⁴+ 3m⁴n²π⁶/l⁶+ 3m²n⁴π⁶/l⁶

+ 4m²n⁶π⁸/l⁸+ 4n²m⁶π⁸/l⁸+ 6n⁴m⁴π⁸/l⁸)ψ²_nm

=

∑∞ n,m=0

+ 4m²n⁶π⁸/l⁸+ 4n²m⁶π⁸/l⁸+ 6n⁴m⁴π⁸/l⁸)

·

( ϕ_nm

1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴ )2

<

∑∞ n,m=0

16(l⁴+n⁸π⁸/l² +m⁸π⁸/l²) ϕ²_nm

1 +n⁸π⁸/l⁸+m⁸π⁸/l⁸

=

∑∞ n,m=0

16ϕ²_nml⁴ <∞,

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and

∥ψ∥²H⁴

=

∑∞ n,m=0

+ 4m²n⁶π⁸/l⁸+ 4n²m⁶π⁸/l⁸+ 6n⁴m⁴π⁸/l⁸)ψ²_nm∥φ_nm∥²L²

<

∑∞ n,m=0

l²(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+n⁶π⁶/l⁶+m⁶π⁶/l⁶ +n⁸π⁸/l⁸+m⁸π⁸/l⁸+ 2m²n²π⁴/l⁴+ 3m⁴n²π⁶/l⁶+ 3m²n⁴π⁶/l⁶

+ 4m²n⁶π⁸/l⁸+ 4n²m⁶π⁸/l⁸+ 6n⁴m⁴π⁸/l⁸)ψ²_nm, ψ ∈H⁴(Ω) follows.

Noting that the solution of (∆² −∆ +I)ψ = ϕ is characterized as the solution of

∫

Ω

∆ψ∆v dx+

∫

Ω

∇ψ· ∇v dx+

∫

Ω

ψv dx=

∫

Ω

ϕv dx, ∀v ∈H²(Ω), we have

∫

Ω

(∆²ψ−∆ψ+ψ)v dx+

∫

∂Ω

∂ψ

∂νv dx+

∫

∂Ω

∂(∆ψ)

∂ν v dx−

∫

∂Ω

∂v

∂ν∆ψ dx=

∫

Ω

ϕv dx, and thus

∫

∂Ω

∂ψ

∂νv dx+

∫

∂Ω

∂(∆ψ)

∂ν v dx−

∫

∂Ω

∂v

∂ν∆ψ dx= 0.

If there exists another solution ˜ψ ̸=ψ, we get (∆²−∆ +I) ˜ψ =ϕ,

which means the existence of a non-trivial solution of the equation (∆² −

∆ +I)φ= 0. But noting that

((∆²−∆ +I)φ, φ)_L²_(Ω)=∥φ∥²_H²_(Ω)

holds, (∆²−∆ +I)φ= 0 derives∥φ∥²_H²_(Ω) = 0, i.e. φ= 0, which contradicts the assumption and the uniqueness of the solution of (∆² −∆ +I)ψ = ϕ follows. 2

Now we derive an estimation for the projection P_N, an imbedding constant from H¹(Ω) toL^p(Ω) for 2≤p <∞ and an imbedding constant from H²(Ω) to L^p(Ω) for 2≤p≤ ∞.

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Lemma 2.0.4 For all z ∈H⁴(Ω), we have

∥z−P_Nz∥H²(Ω) ≤C₂(N)∥△²z− △z+z∥L²(Ω), where C₂(N) = √

1

1+(N+1)²π²/l²+(N+1)⁴π⁴/l⁴. And for z∈H²(Ω),

∥z−PNz∥L²(Ω) ≤C2(N)∥z−PNz∥H²(Ω)

holds.

Proof. For z = ∑^∞

n,m=0

c_nmφ_nm ∈H²(Ω), we obtain

∥z−P_Nz∥²_H²_(Ω)

≤∥z−PNz∥²L²(Ω)+∥(z−PNz)x∥²L²(Ω)+∥(z−PNz)y∥²L²(Ω)+∥(z−PNz)xx∥²L²(Ω)

+∥(z−P_Nz)_yy∥²L²(Ω)+ 2((z−P_Nz)_xx,(z−P_Nz)_yy)_L²_(Ω)

=

∑∞ n=N+1

∑∞ m=0

c²_nm∥φ_nm∥²L²(Ω)+

∑N n=0

∑∞ m=N+1

c²_nm∥φ_nm∥²L²(Ω)

+

∑∞ n=N+1

∑∞ m=0

c²_nm∥(φ_nm)_x∥²_L²_(Ω)+

∑N n=0

∑∞ m=N+1

c²_nm∥(φ_nm)_x∥²_L²_(Ω)

+

∑N n=0

∑∞ m=N+1

c²_nm∥(φ_nm)_y∥²L²(Ω)+

∑∞ n=N+1

∑∞ m=0

c²_nm∥(φ_nm)_y∥²L²(Ω)

+

∑∞ n=N+1

∑∞ m=0

c²_nm∥(φnm)xx∥²L²(Ω)+

∑N n=0

∑∞ m=N+1

c²_nm∥(φnm)xx∥²L²(Ω)

+

∑N n=0

∑∞ m=N+1

c²_nm∥(φ_nm)_yy∥²_L²_(Ω)+

∑∞ n=N+1

∑∞ m=0

c²_nm∥(φ_nm)_yy∥²_L²_(Ω)

+

∑N n=0

∑∞ m=N+1

c²_nm2((φ_nm)_xx,(φ_nm)_yy)_L²_(Ω)+

∑∞ n=N+1

∑∞ m=0

c²_nm2((φ_nm)_xx,(φ_nm)_yy)_L²_(Ω)

=

∑N n=0

∑∞ m=N+1

c²_nm(∥φ_nm∥²_L²_(Ω)+∥(φ_nm)_x∥²_L²_(Ω)+∥(φ_nm)_y∥²_L²_(Ω)+∥(φ_nm)_xx∥²_L²_(Ω) +∥(φ_nm)_yy∥²L²(Ω)+ 2((φ_nm)_xx,(φ_nm)_yy)_L²_(Ω)) +

∑∞ n=N+1

∑∞ m=0

c²_nm(∥φ_nm∥²L²(Ω)

+∥(φ_nm)_x∥²L²(Ω)+∥(φ_nm)_y∥²L²(Ω)+∥(φ_nm)_xx∥²L²(Ω)+∥(φ_nm)_yy∥²L²(Ω)

+ 2((φ_nm)_xx,(φ_nm)_yy)_L²_(Ω))

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≤ ∑^∞

n=N+1

∑∞ m=0

c²_nm(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴)∥φ_nm∥²_L²_(Ω)

+

∑N n=0

∑∞ m=N+1

c²_nm(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴+ 2n²m²π⁴/l⁴)∥φ_nm∥²L²(Ω)

≤ 1

1 + (N + 1)²π²/l²+ (N + 1)⁴π⁴/l⁴

·

∑∞ n=0

∑∞ m=0

(1 +n²π²/l²+m²π²/l²+n⁴π⁴/l⁴+m⁴π⁴/l⁴ + 2n²m²π⁴/l⁴)²c²_nm∥φ_nm∥²L²(Ω)

= 1

1 + (N + 1)²π²/l²+ (N + 1)⁴π⁴/l⁴∥∆²z−∆z+z∥²L²(Ω). So we get C2(N) =

√ 1

1+(N+1)²π²/l²+(N+1)⁴π⁴/l⁴.

For ∥z − P_Nz∥L², we use the so-called Aubin-Nitsche technique. We consider the linear equation:





(∆²−∆ +I)Φ =z−PNz in Ω,

∫

∂Ω

(∂Φ

∂νv−∆Φ∂v

∂ν +∂(∆Φ)

∂ν v )

dx= 0,for ∀v ∈H²(Ω).

By Lemma 2.0.3, we know that there exists the unique solution Φ∈H⁴(Ω).

Therefore, we have

∥z−PNz∥²L²(Ω)

=(z−P_Nz, z−P_Nz)_L²_(Ω)= (z−P_Nz,(∆²−∆ +I)Φ)_L²_(Ω)

=(z−P_Nz,Φ)_L²_(Ω)+ (∇(z−P_Nz),∇Φ)_L²_(Ω)+ (∆(z−P_Nz),∆Φ)_L²_(Ω)

−

∫

∂Ω

(∂Φ

∂ν(z−P_Nz)−∆Φ∂(z−P_Nz)

∂ν + ∂(∆Φ)

∂ν (z−P_Nz) )

dx

=⟨z−P_Nz,Φ⟩H²(Ω)

=⟨z−P_Nz,Φ−P_Nϕ⟩H²(Ω)

≤∥z−P_Nz∥H²(Ω)∥Φ−P_NΦ∥H²(Ω)

≤∥z−PNz∥H²(Ω)C2(N)∥z−PNz∥L²(Ω).

So ∥z−P_Nz∥L²(Ω) ≤C₂(N)∥z−P_Nz∥H²(Ω) holds. 2

Lemma 2.0.5 Ω = (a, b)×(a, b), a < b. Then, for all u∈H¹(Ω), we have

∥u∥L^p(Ω) ≤Kp∥u∥H¹(Ω),(2≤p <∞) where Kp =

( 1

b−a +₂^√^p₂

)(p−2)/p

.

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Proof. For (x, y)∈Ω, we have (x−a)|u(x, y)|^p/2 =∫x

a

∂

∂t[(t−a)|u(t, y)|^p/2]dt

=∫x

a |u(t, y)|^p/2dt+^p₂∫x

a(t−a)|u(t, y)|^p/2⁻²u(t, y)^∂u_∂x(t, y)dt

≤∫x

a |u(t, y)|^p/2dt+ ^p₂(b−a)∫x

a |u(t, y)|^p/2⁻¹^∂u

∂x(t, y)dt, (2.0.2) and

(b−x)|u(x, y)|^p/2 =−∫b x

∂

∂t[(b−t)|u(t, y)|^p/2]dt

=∫_b

x |u(t, y)|^p/2dt− ^p₂∫_b

x(b−t)|u(t, y)|^p/2⁻²u(t, y)^∂u_∂x(t, y)dt

≤∫b

x|u(t, y)|^p/2dt+^p₂(b−a)∫b

x |u(t, y)|^p/2⁻¹^∂u

∂x(t, y)dt.

(2.0.3) Adding (2.0.2) and (2.0.3), we get

(b−a)|u(x, y)|^p/2 ≤∫_b

a |u(t, y)|^p/2dt+^p₂(b−a)∫_b

a |u(t, y)|^p/2⁻¹^∂u

∂x(t, y)dt

=:f(y).

(2.0.4) Analogously, integrating instead in y-direction,

(b−a)|u(x, y)|^p/2 ≤∫b

a |u(x, t)|^p/2dt+^p₂(b−a)∫b

a |u(x, t)|^p/2⁻¹^∂u_∂y(t, y)dt

=:g(x)

(2.0.5) holds.

Multiplying (2.0.4) and (2.0.5), we have

(b−a)²|u(x, y)|^p ≤f(y)g(x) and integrating over Ω we obtain

(b−a)²

∫

Ω

|u(x, y)|^pdxdy

≤

∫

Ω

f(y)g(x)dxdy = (∫ b

a

f(y)dy

) (∫ b a

g(x)dx )

= (∫

Ω

|u(t, y)|^p/2dtdy+ p

2(b−a)

∫

Ω

|u(t, y)|^p/2−1 ∂u

∂x(t, y) dtdy

)

· (∫

Ω

|u(x, t)|^p/2dxdt+p

2(b−a)

∫

Ω

|u(x, t)|^p/2−1 ∂u

∂y(x, t) dxdt

)

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≤



∫

Ω

|u|^p/2dxdy+p

2(b−a) (∫

Ω

|u|^p⁻²dxdy

)1/2(∫

Ω

∂u

∂x

²dxdy

)1/2



·



∫

Ω

|u|^p/2dxdy+p

2(b−a) (∫

Ω

|u|^p⁻²dxdy

)1/2(∫

Ω

∂u

∂y

²dxdy )1/2

 and thus, using (A +Bλ)(A+ Bµ) = A² +AB(λ +µ) +B²λµ ≤ A² +

√2AB√

λ²+µ²+¹₂B²(λ²+µ²) = (A+√¹ 2B√

λ²+µ²)², we have (b−a)²

∫

Ω

|u(x, y)|^pdxdy

≤ (∫

Ω

|u|^p/2dxdy+ p 2√

2(b−a) (∫

Ω

|u|^p⁻²dxdy

)1/2(∫

Ω

|∇u|²dxdy

)1/2)2

i.e. we obtain

∫

Ω

|u|^p ≤ (

1 b−a

∫

Ω

|u|^p/2+ p 2√

2 (∫

Ω

|u|^p⁻²

)1/2(∫

Ω

|∇u|²

)1/2)2

, (2.0.6) here and in the following we omit to write dxdy.

When p= 4, from (2.0.6), we have

∫

Ω|u|⁴ ≤(

b−a1

∫

Ω|u|²+√ 2(∫

Ω|u|²)1/2(∫

Ω|∇u|²)1/2)2

≤(

1 b−a

∫

Ω|u|²+^√₂²(∫

Ω|u|²+∫

Ω|∇u|²))²

≤(

1 b−a +^√₂²

)2(∫

Ω|u|²+∫

Ω|∇u|²)2

= ( 1

b−a+ ^√₂² )2

∥u∥⁴_H¹_(Ω),

therefore, for 2≤p < 4 by using H¨older’s inequality, we get

∫

Ω|u|^p =∫

Ω|u|⁴⁻^p|u|^2p⁻⁴ ≤(∫

Ω|u|²)(4−p)/2(∫

Ω|u|⁴)(p−2)/2

≤ ∥u∥⁴_H⁻¹^p_(Ω)((

1 b−a+ ^√₂²

)2

∥u∥⁴_H¹_(Ω)

)(p−2)/2

≤(

1 b−a +^√₂²

)p−2

∥u∥^p_H1(Ω), that is,

∥u∥L^p(Ω) ≤ (

1 b−a +

√2 2

)(p−2)/p

∥u∥H¹(Ω) (2≤p <4) (2.0.7)

反応拡散方程式系におけるパターン形成に対する計 算機援用証明