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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access April 13, 2017

The quasilinear parabolic kirchhoff equation

  • Łukasz Dawidowski EMAIL logo
From the journal Open Mathematics

Abstract

In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.

MSC 2010: 35K55; 35K15

1 Introduction

G. Kirchhoff in [1] proposed the hyperbolic integro-differential equation in order to describe small, transversal vibrations of an elastic string of length l (at rest) when the longitudinal motion can be considered negligible with respect to the transversal one.

In their papers M. Gobbino [2] and M. Nakao [3] considered some generalized degenerate Kirchhoff equations. M. Gobbino studied the equation:

ut(1+||u||L2(Ω)2)Δu=0,

but his method does not use fixed point theorems and can not be applied to the problem considered in this article. In another papers M. Ghisi and M. Gobbino [4, 5] showed certain connections between the above equation and equation of hyperbolic type containing term utt. However M. Nakao proved the existence of solutions of the equation of hyperbolic type. We will investigate a quasilinear parabolic generalization of the Kirchhoff equation.

The proof of the existence of solution of problem considered in this paper, which is indeed quasilinear (i.e. the derivative of solution is a part of coefficient of the main part), can not be carried out using most classical methods. This paper is devoted to this proof.

Consider the Dirichlet problem for quasilinear generalized degenerate Kirchhoff equation

(1) ut(1+||u||L2(Ω)2)Δu+g(u,x)=0

with initial condition

(2) u(0,x)=u0(x),xΩ,

and boundary condition of the Dirichlet type

(3) u|Ω=0.

We will assume that u0H2(Ω) and Ω ⊆ ℝN is a domain of the class C2.

The following conditions will be imposed on the nonlinear function g: ℝ × Ω → ℝ throughout the paper:

(A1) There exists a function d:Ω → ℝ, such that ∫Ω d(x) dx = d < ∞ and a constant c > 0, that

g(u,x)ucu2+d(x).

(A2) There exists a constant > 0, such that

|g(u,x)|c¯(1+|u|q),

with certain exponent qN+2N .

(A3) There exist constants c1, c2 >0 and exponents s1(0,4N2),s2(0,N+4N2) such that:

| gu |c1(1+|u|s1)and| gxi |c2(1+|u|s2).

In dimension N = 2 we assume only that s1, s2 > 0.

(A4) The function g is locally Lipschitz continuous with respect to the first variable, i.e. there exist constants L > 0, q1,q2(0,NN4) (or if N ≤ 4, then q1, q2 ∈ ℝ), such that

|g(u1,x)g(u2,x)|L|u1u2|(1+|u1|q1+|u2|q2).

(A5) g(0, x) = 0 for all x ∈ Ω.

Remark 1.1

Instead of assuming (A2), (A4) and the first part of (A3) (i.e. there exist constant c1 >0 and s1(0,4N2) such that guc1(1+|u|s1) ) we can assume that:

There exist constant L1 >0 and exponents r1,r2(0,2N) such that

(4) |g(u1,x)g(u2,x)|L1|u1u2|(1+|u1|r1+|u2|r2).

Putting u2 = 0 to (4) and using (A5) we obtain:

|g(u1,x|=|g(u1,x)g(0,x)|L1|u1|(1+|u1|r1).

When we note that index r1 + 1 is no greater than q we observe that assumption (A3) holds. Similarly, as a consequence of (4)

|g(u1,x)g(u2,x)||u1u2|L2(1+|u1|r1+|u2|r2).

Taking the limit with u2u1 we obtain that |gu|L2(1+|u1|r1+|u1|r2) and when we put s1 := max(r1, r2), then (A4) holds.

Constructing a solution of (1) the Leray – Schauder Principle will be used (see e.g. [6], p. 189). We recall it here for completeness of the presentation:

Proposition 1.2

(Leray – Schauder Principle). Consider a transformation y = T(x, k) where x, y belong to a Banach space X and k is a real parameter which varies in a bounded interval, say akb. Assume that

  1. T(x, k) is defined for all xX and akb,

  2. for any fixed k, T(x, k) is continuous as a function of x, i.e. for any x0X and for any ε > 0 there exists a δ > 0 such that ||T(x, k) − T(x0, k)|| < ε if ||xx0|| < δ,

  3. for x varying in bounded set in X, T(x, k) is uniformly continuous in k, i.e. for any bounded set X0X and for any ε > 0 there exists a δ > 0 such that if xX0, |k1k2| < δ, ak1, k2b, thenT(x, k1) − T(x, k2)∥ < ε

  4. for any fixed k, T(x, k) is a compact transformation, i.e. it maps bounded subsets of X into compact subsets of X,

  5. there exists a (finite) constant M such that every possible solution X of xT(x, k) = 0 (xX, akb) satisfies: ∥x∥ < M,

  6. the equation xT(x, a) = y has the unique solution for any yX.

    Then there exists a solution of the equation xT(x, b) = 0.

Assumption (f) means that Leray-Schauder degree

degLS(IT(,a);B(0,M);0)0

with the constant M which comes from assumption (e). The more standard version of Leray-Schauder Principle, called also Leray-Schauder continuation theorem, can be found e.g. in [7], p. 351 (Theorem 13.3.7).

2 Main theorem

Let us fix arbitrary T > 0.

We introduce an operator F : L ( [ 0 , T ] , H 0 1 ( Ω ) ) × [ 0 , 1 ] L ( [ 0 , T ] , H 0 1 ( Ω ) ) in such a way, that for every function υ L ( [ 0 , T ] ) , H 0 1 ( Ω ) ) and α ∈ [0, 1], u = F(υ, α) is a solution of the equation

(5) ut(1+αυL2(Ω)2)Δu+g(u,x)=0,

with initial – boundary conditions:

u(0,x)=u0(x),xΩ,u|Ω=0.

We will search a fixed point of the operator F(·, 1) in L([0,T],H01(Ω)) .

The existence of the solution of the problem (1) is equivalent to the existence of the fixed point of operator F(·, 1) in L([0,T],H01(Ω)) .

We have:

Theorem 2.1

Under the assumptions (A1), (A2), (A3), (A4) and (A5) there exists a solution of the problem (1) with initial-boundary conditions (2), (3) in the space L([0,T],H01(Ω)) .

It can be seen that, using standard theory (see e.g. [8], chapter 3 for details), for α = 0 the equation uF(u, 0) = y has a unique solution for any yL([0,T]),H01(Ω)) ; equivalently, the semilinear heat equation

utΔu+g(u,x)=0

with Dirichlet boundary condition has a unique solution.

The proof of the theorem will be given in a few steps. We start with obtaining certain a priori estimates.

3 Some lemmas

First it can be mentioned that when u0H2(Ω) then, using the method of Tanabe and Sobolevski (see [9], page 438), the solution of the problem (5) varies in the space H2(Ω).

Lemma 3.1

There exists A ∈ ℝ that for all t ∈ (0, T) this estimate holds:

Ω u 2 d x e 2 A t Ω u 0 2 d x + d A d A .

Proof

Multiplying the equation (5) by u and integrating over Ω we obtain:

Ωuutdx(1+αυL2(Ω)2ΩΔuudx+Ωg(u,x)udx=0.

Then from (A1):

12ddtΩ|u|2dx(1+αυL2(Ω)2)ΩΔuu+Ω(cu2+d(x))dx.

Integrating first right hand side component by parts we obtain:

12ddtΩ|u|2dx(1+αυL2(Ω)2)Ω|u|2dx+cΩu2dx+d.

Using the Poincaré inequality ∫Ω |u|2 dxpΩ |∇u|2 dx, we have next:

12ddtΩ|u|2dx1p(1+αυL2(Ω)2)Ω|u|2dx+xΩu2dx+d,

so that

12ddtΩ|u|2dx[c1p(1+αυL2(Ω)2)]Ω|u|2dx+d.

Choosing A=c1p the estimates holds

12ddtΩ|u|2dxAΩ|u|2dx+d.

Finally using Gronwall inequality (see [10], p. 35):

Ωu2dxe2At(Ωu02dx+dA)dA,

for t ∈ (0, T). □

Lemma 3.2

There exist constants B, D ∈ ℝ such that:

Ω|u|2dxeBt(Ω|u0|2dxDB)+DB.

Proof

Multiplying equation (5) by Δu and integrating over Ω:

ΩΔuutdx(1+αυL2(Ω)2)ΩΔuΔudx+Ωg(u,x)Δudx.

Integrating by parts:

ΩΔuutdx=12ddtΩ|u|2dx.

Then using Cauchy inequality with ε=12 and (A2):

12ddtΩ|u|2dx=(1+αυL2(Ω)2)ΩΔuΔudx+Ωg(u,x)Δudx(1+αυL2(Ω)2)ΩΔuΔudx+12Ω(g(u,x))2dx+12Ω(Δu)2dx(12+αυL2(Ω)2)ΩΔuΔudx+12c¯2Ω(1+|u|q)2dx.

and

12ddtΩ|u|2dx(12+αυL2(Ω)2)ΩΔuΔudx+12c¯2Ω(1+|u|q)2dx(12+αυL2(Ω)2)Ω|Δu|2dx+c¯2Ω(1+|u|2q)dx==(12+αυL2(Ω)2)Ω|Δu|2dx+c¯2|Ω|+c¯2Ω|u|2qdx.

Since q<N+2N when θ=N4(11q) , the following estimate holds:

ϕL2q(Ω)2qc2ϕL2(Ω)2q(1θ)ϕH2(Ω)2qθand2qθ<1.

Consequently, our resulting estimate has the form:

12ddtΩ|u|2dx(12+αυL2(Ω)2)Ω(Δu)2dx+c¯2|Ω|+c¯2c2uL2(Ω)2q(1θ)uH2(Ω)2qθ

Because the norms ∥ · ∥H2(Ω) and ∫Ω ·2 dx + ∫Ω(Δ·)2 dx are equivalent on the domain of (−Δ) (for more details see e.g. [11]):

(12+αυL2(Ω)2)Ω(Δϕ)2dxh1ϕH2(Ω)2h2Ωϕ2dx,

for some constants h1, h2 > 0. Thus

12ddtΩ|u|2dx[h1uH2(Ω)2h2Ωu2dx]c¯2|Ω|+c¯2c2uL2(Ω)2q(1θ)uH2(Ω)2qθ==h1uH2(Ω)2+c¯2c2uL2(Ω)2q(1θ)uH2(Ω)2qθ+c¯2|Ω|+h2Ωu2dx.

Due to lemma 3.1, supt[0,T]Ωu2dxe=e(c,d,T) , so that the estimate holds

12ddtΩ|u|2dxh1uH2(Ω)2+c¯2c2uL2(Ω)2q(1θ)+c¯2|Ω|+h2e.

Using the Young inequality with ϵ1

c¯2c2uL2(Ω)2q(1θ)uH2(Ω)2qθϵ1h1uH2(Ω)2+auL2(Ω)P,

with a positive constant a = a(, c2, ϵ1), and the exponent P=11qθ is chosen in such a way that Young inequality holds. Then:

12ddtΩ|u|2dx(ϵ1+1)h1uH2(Ω)2+auL2(Ω)P+c¯2|Ω|+h2e.

Since there exists a constant č >0 such that u H 2 ( Ω ) 2 C ˇ Ω u 2 d x and uL2(Ω)2b< then:

1 2 d d t Ω u 2 d x ( ϵ 1 + 1 ) h 1 c ˇ Ω u 2 d x + a b P + c 2 Ω + h 2 e .

Therefore

d d t Ω u 2 d x + 2 ( ϵ 1 + 1 ) h 1 c ˇ Ω u 2 d x 2 a b P + 2 c ¯ 2 Ω + 2 h 2 e

and the right side is a constant. Using Gronwall inequality (see [10], p. 35), denoting B = 2(ϵ1 + 1)h1č and D = 2abP + 22|Ω| + 2h2e, we obtain:

Ω|u|2dxeBt(Ω|u0|2dxDB)+DB.

 □

Remark 3.3

The two previous lemmas provide us an a priori estimate of the solution u of (5) in the space([0, T],H1(Ω)).

Let us take a constant M > 0 such that:

uL([0,T],H1(Ω))=supt[0,T]u(t)H1(Ω)<M

The lemmas show also that, if u is a fixed point of the operator F, its normu([0,T],H1(Ω)) will be bounded by M, since the constants in both lemmas are independent of u and α.

Finally a third a priori estimation in H2(Ω) will be shown:

Lemma 3.4

There exists a constant M1 > 0 such that

ΔuL2(Ω)M1<,

Proof

By applying the Laplace operator Δ to (5), multiplying the result by Δu and integrating over Ω we obtain:

ΩΔuΔutdx(1+αυL2(Ω)2ΩΔuΔ2udx+Ωg(u,x)Δ2udx=0.

Integrating by parts and using (A5):

12ddtΩ|Δu|2dx+(1+αυL2(Ω)2Ω|Δu|2dx=Ωg(u,x)(Δu)dx.

Then thanks to the Cauchy inequality

(6) 12ddtΩ|Δu|2dx+(12+αυL2(Ω)2)Ω|Δu|2dx12Ω|g(u,x)|2dx.

Now, using assumption (A3), we will estimate last integral:

(7) 12Ω|g(u,x)|2dxΩi=1N(gu)2(uxi)2dx+Ωi=1N(gxi)2dxi=1NΩ(gu)4dxΩ(uxi)4dx+i=1NΩ(gxi)2dxNc1Ω(1+|u|4s1)dxuW1,4(Ω)2+Nc2Ω(1+|u|2s2)dx(c1N|Ω|+c1NuL4s1(Ω)2s1)uW1,4(Ω)2+c2N|Ω|+c2N|u|L2s2(Ω)2s2

Next the norms ∥ · ∥L4s1(Ω), ∥ · ∥w1,4(Ω) and ∥ · ∥L2s2(Ω) are estimated using Gagliardo-Nierenberg inequality. Since s1(0,4N2),s2(0,N+4N2) it is possible to find constants c3, c4, c5 > 0, and powers θ1, θ2, θ3 ∈ (0,1), such that:

uL4s1(Ω)c3uH3(Ω)θ1uH1(Ω)1θ1,uW1,4(Ω)c4uH3(Ω)θ2uH1(Ω)1θ2,2s1θ1+2θ2<2,uL2s2(Ω)c5uH3(Ω)θ3uH1(Ω)1θ3,2s2θ3<2.

Due to Remark 3.3, ∥uH1(Ω) < M, inequality (7) will take the form:

(8) 1 2 Ω g ( u , x ) 2 d x c 1 N Ω + c 1 N u L 4 s 1 ( Ω ) 2 s 1 u W 1 , 4 ( Ω ) 2 + + c 2 N Ω + c 2 N u L 2 s 2 ( Ω ) 2 s 2 c 1 N Ω + c 1 N c 3 M 2 s 1 ( 1 θ 1 ) u H 3 ( Ω ) 2 s 1 θ 1 c 4 M 2 ( 1 θ 2 ) u H 3 ( Ω ) 2 θ 2 + + c 2 N Ω + c 2 N c 5 M 2 s 2 ( 1 θ 3 ) u H 3 ( Ω ) 2 s 2 θ 3 .

Choosing θ = max(2s1θ1 + 2θ2, 2s2θ3) we find that θ < 2. Then, defining

const=c2N|Ω|

and

k=max(c1N|Ω|c4M2(1θ2),c1Nc3M2s1(1θ1)c4M2(1θ2),c2Nc5M2s2(1θ3),

we obtain

12Ω|g(u,x)|2dxconst+kuH3(Ω)θ.

Estimate (6) will be extended to:

12ddtΩ|Δu|2dx+(12+αυL2(Ω)2)Ω|Δu|2dxconst+kuH3(Ω)θ.

Now, since the norms ∥ · ∥H3(Ω) and ∥ · ∥L2(Ω) + ∥∇ · ∥L2(Ω) + ∥ Δ · ∥L2(Ω) + ∥∇Δ · ∥L2(Ω) are equivalent, and due to remark 3.3, we have:

c¯>0uH3(Ω)c¯(M+ΔuL2(Ω)).

Since θ < 2, we can find a constant c^<12 such that:

kuH3(Ω)θc^ΔuL2(Ω)2+const=c^Ω|Δu|2dx+const

Then we have:

ddtΩ|Δu|2dx+αυL2(Ω)2ΩΔudxconst,

Now, using the Calderon-Zygmund type inequality (see [12], pp. 186-187):

c¯>0ϕD(Δ3/2)Ω|Δϕ|2dxc˜Ω|Δϕ|2dx.

we can find positive constants h1, h0 such that:

ddtΩ|Δu|2dx+h1Ω|Δu|2dxh2,

Using the Gronwall inequality (see [10], p. 35) we finally obtain:

Ω|Δu|2dxeh1t(Ω(Δu0)2dxh2h1)+h2h1

for t ∈ (0, T). Then defining M1=supt(0,T)[eh1t(Ω(Δu0)2dxh2h1)+h2h1] the proof will be completed. □

Remark 3.5

Above lemmas show that every eventual solution in the sense of Theorem 2.1 has to be an element of the space L([0,T],H2(Ω)H01(Ω)) .

4 Proof of the main theorem

This section is devoted to the proof of the Theorem 2.1. Three conditions from the Leray – Schauder Principle: continuities (b), (c) and compactness (d) will be verified.

It has to be proved that the operator F(·, α) is compact, i.e. it maps bounded subsets of L([0,T],H01(Ω)) into compact subsets of L([0,T],H01(Ω)) . Let us fix α ∈ [0,1]. If the bounded subset A of L([0,T],H01(Ω)) is taken, then F(A, α) is bounded in the space L([0,T],H2(Ω)H01(Ω)) . Using the equation (5), we can write:

ut=(1+αυL2(Ω)2)Δug(u,x)

and because an element υA, ΔuL2(Ω) and thanks to (A2) the function gL2(Ω), we can deduce that utL2(Ω). Additionally, the embedding H2(Ω) ⊆ H1(Ω) is compact and H1(Ω) ⊆ L2(Ω) is continuous. Using Aubin lemma (see e.g. [13] and [14]) the set of values of operator F(A, α) is compact in L([0,T],H01(Ω)) .

Now we prove that for any fixed α ∈ [0, 1], the operator F(υ, α) is continuous as a function of υ, i.e. for any υ1L([0,T],H01(Ω)) and for any ε > 0 there exists a δ > 0 such that ∥F(υ1, α) − F(υ2, α)∥ < ε if ∥υ1v2∥ < δ.

Let us take α ∈ [0,1] and υ1,υ2L([0,T],H01(Ω)) . Let u1,u2L([0,T],H01(Ω)) be the values of the operator F corresponding to υ1 and υ2, i.e.

( u 1 ) t ( 1 + α υ 1 L 2 ( Ω ) 2 ) Δ u 1 + g ( u 1 , x ) = 0 , ( u 2 ) t ( 1 + α υ 2 L 2 ( Ω ) 2 ) Δ u 2 + g ( u 2 , x ) = 0.

Subtracting the above equations, defining u: = u1u2, using (A4), it can be seen that:

(9) ut(1+αυ1L2(Ω)2)Δuα(Δu2)υ1υ2L([0,T],H1(Ω))+Lu(1+|u1|q1+|u2|q2).

Then we have that u(0, x) = û0 = 0 for all x ∈ Ω. Analogously as in the lemmas above, we see that for all ε > 0 there exists some δ > 0 such that

υ1υ2L([0,T],H01(Ω))<δu1u2L([0,T],H1(Ω))<ε.

As an example we prove the estimation for ∥uL2(Ω). Similarly, we can prove the estimations for ∥∇uL2(Ω), ∥ΔuL2(Ω).

Lemma 4.1

If function uL([0,T],H01(Ω)) is a solution of inequality (9) with initial condition u(0, x) = 0 for x ∈ Ω, then for all ε > 0 there exists some δ > 0 such that if υ1υ2L([0,T],H01(Ω))<δ then

uL([0,T],L2(Ω))=u1u2L([0,T],L2(Ω))<ε.

Proof

The proof is similar to the proof of Lemma 3.1. By multiplying the inequality (9) by |u| and integrating over Ω we obtain:

Ω|u|utdx(1+αυL2(Ω)2)ΩΔu|u|dxαυ1υ2L([0,T],H1(Ω))Ω|u|(Δu2)dx+LΩu2(1+|u1|q1+|u2|q2)dx.

Then using Cauchy inequality and statement of Lemma 3.4 we obtain:

Ω|u|(Δu2)dx12Ωu2dx+12Ω(Δu2)2dx12Ωu2dx+12M2.

Analogously, using (A4) and the fact that H2(Ω) ⊆ L2q1(Ω) ∩ L2q2(Ω):

Ω u 2 ( 1 + u 1 q 1 + u 2 q 2 ) d x 1 2 ϵ Ω u 2 d x + 2 ϵ Ω ( 1 + u 1 2 q 1 + u 2 2 q 2 ) d x 1 2 ϵ Ω u 2 d x + 2 ϵ ( Ω + 2 M 2 ) 1 2 Ω u 2 d x + ϵ C 1 .

for some ϵ > 0. Integrating by parts component ∫Ω · |u| dx we obtain:

1 2 d d t Ω u 2 d x + ( 1 + α υ L 2 ( Ω ) 2 ) Ω u 2 d x α υ 1 υ 2 L ( [ 0 , T ] , H 1 ( Ω ) ) 1 2 Ω u 2 d x + 1 2 M 2 + 1 2 ϵ L Ω u 2 d x + L ϵ C 1 .

Using the Poincaré inequality ∫Ω|u|2 dxpΩ |∇u|2 dx, we have next:

1 2 d d t Ω u 2 d x + 1 p ( 1 + α υ L 2 ( Ω ) 2 ) Ω u 2 d x α υ 1 υ 2 L ( [ 0 , T ] , H 1 ( Ω ) ) 1 2 Ω u 2 d x + 1 2 M 2 + 1 2 ϵ L Ω u 2 d x + L ϵ C 1 .

so that

d d t Ω u 2 d x C ^ Ω u 2 d x + C ^ 1 .

where

C^=[αυ1υ2L([0,T],H1(Ω))+Lϵ2p(1+αυL2(Ω)2)]

and

C ^ 1 = α υ 1 υ 2 L ( [ 0 , T ] , H 1 ( Ω ) ) M 2 + L ϵ C 1 .

Finally, using the Gronwall inequality (see [10], p. 35):

Ωu2dxeC^t(Ωu^02dx+C^1C^)C^1C^,

for t ∈ (0, T). Noting that û0 = 0, we obtain:

Ωu2dxC^1C^eC^tC^1C^,

Let us fix ε > 0 and take δ > 0 and ϵ > 0 such that

υ1υ2L([0,T],H1(Ω))<δC^1=αυ1υ2L([0,T],H1(Ω))M2+LϵC1<δδC^supt(0,T)eC^tδC^<ε.

Then:

Ωu2dxδC^supt(0,T)eC^tδC^<ε.

 □

Continuity of the operator F(υ, α) with respect to the parameter α will be verified in the similar way. Let XL([0,T],H01(Ω)) be a bounded subset and υX. Then there exists a constant N > 0 such that

υL([0,T],H01(Ω))N,forυX

Let us take α1, α2 ∈ [0,1] and assume that u1, u2 will be solutions of the problem (5), i.e.

( u 1 ) t ( 1 + α 1 υ L 2 ( Ω ) 2 ) Δ u 1 + g ( u 1 , x ) = 0 , ( u 2 ) t ( 1 + α 2 υ L 2 ( Ω ) 2 ) Δ u 2 + g ( u 2 , x ) = 0.

Then, after subtracting equations and defining u = u1u2, we receive

ut(1+α2υL2(Ω)2)ΔuN(Δu2)|α1α2|+Lu(1+|u1|q1+|u2|q2).

Because |α1α2| < δ, the continuity of the operator F with respect to a will be obtained in a similar way as above.

Remark 4.2

As a result of Theorem 2.1 and Remark 3.5 there exists a solution u ∈ ℒ([0, T], H2(Ω)) of the problem (1) with initial-boundary conditions (2), (3).

In this paper the existence of a solution of some quasilinear parabolic generalization of the Kirchhoff equation of the form (1) with initial – boundary condition was proved. Using the Leray-Schauder Principle we obtain the solution in the space L([0,T],H2(Ω)H01(Ω)) for each arbitrary T > 0. Its higher regularity will be studied next.

Acknowledgement

The author is grateful to the referee for valuable remarks improving the original version of the paper.

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Received: 2015-5-11
Accepted: 2015-11-3
Published Online: 2017-4-13

© 2017 Dawidowski

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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