A class of nonlocal nonlinear boundary value problems with definite ...

A class of nonlocal nonlinear
boundary value problems
with definite integrals
Shoji Yotsutani (Ryukoku University)
We have been interested in complete global bifurcation structures of several
nonlocal nonlinear boundary problems arising in various fileds. We show
four typical examples.
1. Oseen’s spiral flow
The first problem is related with the Oseen’s spiral flow [12]. Find a
function U(x) such that
(O)
⎧
Uxx + AU− U
⎪⎨
⎪⎩
2 + C =0,
C :=
x ∈ (−π, π),
1
Z π
U(x)
2π −π
2 dx,
U(−π) =U(π),
Z π
U(x)dx =0
Ux(−π) =Ux(π),
−π
for arbitrarily fixed A.
It is easily seen that U ≡ 0 is the trivial solution of the above problem
for any fixed A. Okamoto [11] started to investigate the global bifurcation
structure of this problem. Moreover, Ikeda-Mimura-Okamoto [4] obtained
the asymptotic shape of solutions as A →−∞.
Let us recall the standard notation of complete elliptic integrals:
K(k) :=
E(k) :=
Z π
2
0
Z π
2
0
1
p 1 − k 2 sin 2 ϕ dϕ, k ∈ [0, 1),
q
1 − k 2 sin 2 ϕ dϕ, k ∈ [0, 1).
Jacobi’s elliptic functions sn(x, k)andcn(x, k) with the modulus k are defined
as follows:
sn −1 Z z
1
(z,k) := p dξ, z ∈ [0, 1], k∈ [0, 1),
(1 − ξ2 )(1 − k2ξ 2 )
0
1

and
We note that
cn 2 (z, k) =1− sn 2 (z,k).
E(0) = K(0) = π
1
, E(1) = 1, K(k) ∼
2 2 log
µ
16
1 − k2
as k → 1.
Ikeda-Kondo-Okamoto-Yotsutani [3] have parameterized all solutions (A, U)
of (O) in terms of the elliptic functions, and clarified the global bifurcation
structure by the following Theorems 1 and 2.
Theorem 1 All the solution (A, U) of (O) are parameterized by
where
{(n 2 A(k),n 2 U(nx − x0; A(k)) : 0

The structure of solutions is similar to that of Oseen’s spiral flow, though
the analysis is more difficult. Kosugi-Morita-Yotsutani [5] have clarified
the global bifurcation structure of this problem. (See, also Kosugi-Morita-
Yotsutani [6].)
We briefly explain about the original equation. Consider the following
Ginzburg-Landau equation:
½
ψxx + λ(1 − |ψ| 2 )ψ =0, x ∈ (−π, π)
ψ(−π) =ψ(π), ψx(−π) =ψx(π).
We here assume that |ψ| > 0andψ is written as the form
ψ = u(x)exp(iθ(x)),
where u and θ are both real-valued smooth functions. Clearly the equation
is equivalent the following system:
⎧
⎪⎨
⎪⎩
uxx − θx 2 u + λ(1 − u 2 )u =0, x ∈ (−π, π),
(u 2 θx)x =0,x∈ (−π, π),
U(−π) =U(π), Ux(−π) =Ux(π),
θ(π) − θ(−π)u =2mπ, θx(−π) =θx(π),
where m is an integer. Thus, θx = C/u 2 for a constant C and hence we
obtain (P).
3. Minimum energy curve
The third problem is related to find the minimum energy curve for given
the length L and area M, which K.Watanabe [14] started to investigate.
For given L>0andM>0withL2−4πM >0, find a function κ(s)
such that
⎧ ©
κss +
⎪⎨
(E)
⎪⎩
1
2κ3 + μκ ª
=0,s∈ [0,L],
s
1 μ := L2 n
M −4πM
R L
0 κ(s)3ds − L
R L
2 0 κ(s)2 o
ds ,
κ(0) = κ(L), κs(0) = κs(L),
κ(s)ds =2π.
R L
0
Murai-Matsumoto-Yotsutani [10] have clarified the global bifurcation strucure
of this problem, though terribly complicated calculations are needed.
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4. Cross-diffusion
The final problem is a limiting equation for the Shigesada-Kawasaki-
Teramoto model with cross-diffusion [13]. This problem is the hardest.
Find (v(x), τ) such that τ > 0, and
⎧ Z 1 µ
τ
τ
a1 − b1 − c1v(x) dx =0,
⎪⎨ 0 v(x) ³ v(x)
τ
´
(C) d2vxx + v · a2 − b2 − c2v =0 in(0, 1),
v
⎪⎩
vx(0) = 0, vx(1) = 0,
v>0 on [0, 1].
where a1,a2,b1,b2,c1,c2,d2 are given positive constants.
We briefly explain the original equation. In 1979, Kawasaki—Shigesada—
Teramoto proposed a cross-diffusion system
⎧
⎪⎨
⎪⎩
ut = {(d1 + ρ12v)u} xx + u(a1 − b1u − c1v) (0

Conjecture 1: Suppose that B

Conjecture 3: Suppose that C>7B/3. There exists the only one
connected non-empty open set D such that (S) has exactly two solutions
(v(x), τ) if and only if d2 ∈ D.
Stability:
C>7B/3, a2 = b2 = c2 =1
C>7B/3, a2 = b2 = c2 =1
Lou-Ni-Yotsutani [8] have almost clarified the existence and the shape of
solutionsasfollows.
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Theorem 3 (Existence) Suppose that B

Theorem 6 (Shape of solutions as d2 → 0 for A < (B +3C)/4) Let
(v(x, d2), τ(d2)) be solutions of (S). IfA0,
C − B
τ(d2)
v(0; d2)
τ(d2)
v(x; d2)
a2
→ ·
2c2
C − A
C − B
→ a2
b2
· C − A
A − B
·
− A,
B+3C
4
for x>0,
C − B
Theorem 7 (Shape of solutions as d2 → 0 for A ≥ (B +3C)/4) Let
(v(x, d2), τ(d2)) be solutions of (S). IfB0,
for x>0,

Now, we will discuss about the uniqueness and non-uniquess. The following
result are a part of joint projects with W.-M. Ni.
Theorem 8 Suppose that B7B/3. There exists an open set D such that
(S) has at least two solutions (v(x), τ) for d2 ∈ D.
References
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Shigesada-Kawasaki-Teramotomodel with weak cross-diffusion, Discrete
Contin. Dyn. Syst. 9 (2003), 1193-1200.
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for Shigesada-Kawasaki-Teramotomodel with strongly coupled crossdiffusion,
Discrete Contin. Dyn. Syst. 10 (2004), 719-730.
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branches of the solutions to a nonlocal boundary-value problem arising in
Oseen’s spriral flows, Commun. Pure Appl. Anal., 3 (2003), 381-390.
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arising in Oseen’s spiralflows, Japan J. Indust. Appl. Math. 18 (2001),
393-403.
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[5] S. Kosugi, Y. Morita and S. Yotsutani, A complete bifurcation diagram
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Pure Appl. Anal., 4 (2005), no.3, 665-682.
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of a 1-d Ginzburg-Landau model, Journal of Mathematical Physics, 46
095111 (2005), no.1, 1-24.
[7] Y. Lou and W.-M. Ni, Diffusion vs. cross-diffusion: An elliptic approach,
J. Differential Equations 154 (1999), 157-190.
[8] Y.Lou,W.-M.NiandS.Yotsutani,On a limiting system in the Lotka—
Volteracompetition with cross-diffusion, Discrete Contin. Dyn. Syst. 10
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I., Arkiv Mat. Astr. Fysik, 20 (1927—1928), No. 14, pp. 1-14: ibid.
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