2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i

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1 1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x, t), h = h(x, t) D a, D h, µ, ν, c, c, ρ 0 ρ(x) ρ (x) Gierer Meinhardt ρ ρ (GM) Ω N Euclid R N Ω (d (a) i,j (x)) (d (h) i,j (x)) N N d a, d h ξ 2 N i,j=1 d (a) i,j ξ iξ j d a ξ 2, ξ 2 N i,j=1 d (h) i,j ξ iξ j d h ξ 2 ξ R N x Ω d (a) i,j (x) d(h) i,j (x) d (a) i,j (x) Ω Hölder (x), d(h) i,j

2 2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i d (h) ν = (ν 1,..., ν N ) Ω Gierer Meinhardt (1.1) i,j=1 x j A t = ε2 Λ a A µ a (x)a + ρ a (A, H, x) Ap H q + σ a(x) (x Ω, t > 0), τ H t = DΛ hh µ h (x)h + ρ h (A, H, x) Ar H s + σ h(x) (x Ω, t > 0). (1.2) (1.3) B a A = 0 B h H = 0 (x Ω, t > 0), A(x, 0) = A 0 (x), H(x, 0) = H 0 (x) (x Ω), ε, D τ µ a (x), µ h (x) Ω Hölder (1.4) 0 < k (a) 1 µ a (x) k (h) 2, 0 < k(h) 1 µ h (x) k (h) 2 (x Ω); ρ a (A, H, x) ρ h (A, H, x) < A < +, < H < +, x Ω (A, H) x Ω Hölder c a, C a, c h, C h 0 < c a ρ a (A, H, x) C a, ρ a (A, H, x) A + ρ a (1.5) (A, H, x) H C a 0 < c h ρ h (A, H, x) C h, ρ h (A, H, x) A + ρ h (1.6) (A, H, x) H C h A 0, H 0, x Ω σ a ρ h (1.7) σ a, σ h C γ (Ω) σ a (x) 0, σ h (x) 0 (x Ω)

3 (A) 3 (1.8) A 0, H 0 C 2+γ (Ω), B a A 0 Ω = B h H 0 Ω = 0 A 0 (x) > 0, H 0 (x) > 0 (x Ω) 0 < γ < 1 p > 0, q > 0, r > 0, s 0 (1.9) 0 < p 1 r < q s + 1 µ a, µ h, ρ a, ρ h Koch-Meinhardt [3]

4 4 (March 13, 2010) 1.2. (1.1) (1.3) Λ a Λ h Laplace = N j=1 2 / x 2 j µ a, µ h [14], [6], [20], [4], [2] min σ a (x) > 0, σ h (x) 0 (p 1)/r < 2/(N + 2) [6] t > 0 t + {(A(x, t), H(x, t)) R 2 x Ω} ρ a (A, H, x) ρ h (A, H, x) 1, ρ h (x) 0 Li, Chen, Qin[4] p 1 < r min σ a (x) > 0 t > 0 Jiang[2] [15] min σ a (x) = 0 (1.10) p 1 < r A C [6], [4], [2], [15] (1.1) (1.3) A. (1.9) (1.10) (1.7) max σ a (x) > 0 (1.1) (1.3) t > 0 (A 0 (x), H 0 (x)) r a, r h, R a, R h r a lim inf min t + r h lim inf min t + A(x, t) lim sup t + max A(x, t) R a, H(x, t) lim sup max R h. t +

5 (A) 5 B. (1.9) (1.10) σ a (x) 0 max σ h (x) > 0 (1.1) (1.3) t > 0 (A 0 (x), H 0 (x)) r h, R a, R h e k(a) 2 t min r h lim inf min t + A 0 (x) A(x, t) (x Ω, t > 0), lim sup t + H(x, t) lim sup t + max H(x, t) R h max A(x, t) R a C. (1.9) (1.10) σ a (x) 0 σ h (x) 0 (1.1) (1.3) t > 0 p, q, r, s, τ, k (a) 1, k(h) 1, C a c h λ µ (A 0 (x), H 0 (x)) C e k(a) 2 t min A 0 (x) A(x, t) Ce λt, e k(h) 2 t min H 0 (x) H(x, t) Ce µt t > 0 x Ω A C p 1 < r d (a) ij = d (h) ij = δ ij, µ a µ h 1, ρ a ρ h 1, σ a, σ h (1.1) (1.2) [4] [7] p 1 > r (1.1) (1.3) p 1 r, q s + 1 s + 1 < (p 1)τ q (1.1) (1.3) σ a (x) σ h (x) 0 (1.1) (1.3) t > 0

6 6 (March 13, 2010) ([7]) A σ a

7 (A) 7 3. σ a (x) σ h (x) 0 : 0 morpho/activities/results.html (1.1) (1.3) Σ a,ε (x), Σ h,d (x) : { ε 2 Λ a Σ a,ε µ a (x)σ a,ε + σ a (x) = 0 (x Ω), (1.11) B a Σ a,ε = 0 (x Ω), { ε 2 Λ h Σ h,d µ h (x)σ h,d + σ h (x) = 0 (x Ω), (1.12) B h Σ h,d = 0 (x Ω). : I: σ a (x) 0 σ h (x) 0; II: σ a (x) 0 σ h (x) 0; III: σ a (x) 0 σ h (x) 0; IV: σ a (x) 0 σ h (x) 0; Wu-Li [20] 1. (, I II) τ > qk (h) 2 /[(p 1)k(a) 1 ] (1.13) ( q min H 0 (x)) > C a (p 1) k (a) 1 (p 1) qk(h) 2 /τ σ a (x) 0 τ ( max A 0 (x) (1.1) (1.3) (A(x, t), H(x, t)) 0 < max A(x, t) Ce k(a) 1 t, max ) p 1 H(x, t) Σ h,d (x) Ce k(h) 1 t/τ C (A 0 (x), H 0 (x)) Σ h,d (x) (1.12)

8 8 (March 13, 2010) (A 0 (x), H 0 (x)) (0, Σ h,d (x)) 1 II 2. ( II) σ a (x) 0 max (x) > 0 S m = min Σ h,d (x), S M = max Σ h,d (x) (A 0 (x), H 0 (x)) (1.14) min {( (S m /S M ) k(h) 2 /k (h) 1 min 0 < max A(x, t) Ce k(a) q ( ) } q H 0 (x)), S m k (h) 1 /k(h) 2 1 t, max > C ( a k (a) 1 max A 0 (x) H(x, t) Σ h,d (x) Ce k(h) 1 t/τ ) p 1 C (A 0 (x), H 0 (x)) Σ h,d (x) (1.12) I τ 3. σ a (x) σ h (x) 0 (1.1) (1.3) (A(x, t), H(x, t)) (i) x Ω t > 0 H(x, t) q > ρ a (A(x, t), H(x, t), x)a(x, t) p 1 /µ a (x) (ii) t + (A(x, t), H(x, t)) (0, 0) τ qk (h) 1 /[k(a) 2 (p 1)] III, IV 4. (1.1) (1.3) (A(x), H(x)) x Ω ρ a (A(x), H(x), x) A(x)p 1 H(x) q < µ a (x) ρ h (A(x), H(x), x) A(x)r H(x) s+1 < µ h(x) :

9 (A) 9 5. ( ) max σ a (x) > 0 max σ h (x) > 0 0 < r < 1 min σ a (x) γ a (max σ a (x)) p γ a σ a (x) m 0 max σ a (x) m 0 (1.1) (1.3) (A (x), H (x)) ( p ( (1.15) A Σ a,ε L C max σ a (x)), H Σ h,d L C max σ a (x) C Σ a,ε, Σ h,d (1.11), (1.12) ) r III κ a, K a S m = min Σ h,d (x) 0 < κ a < K a k (a) 1 ξ + C a S q m ξ p + max σ a (x) = 0 max σ a (x) m 0 m 0 1.5, max σ a (x) 0 κ a = 1 ( (1.16) max σ a (x) + O (1.17) K a = k (a) 1 ( k (a) 1 Sq m C a ) 1/(p 1) k(a) 1 ) (max σ a (x)) p, p 1 max σ a (x) + o(max σ a (x)) 1.6. ( III) (A 0 (x), H 0 (x)) 1.5 max < K a, H 0 (x) max Σ h,d (x) (A 0, H 0 ) C γ A(x, t) A (x) + H(x, t) H (x) Ce γt x Ω t > 0 (A, H ) 1.5

10 10 (March 13, 2010) 1.7. i) 1.3 I t + (0, 0) (1.1) (1.3) τ k (h) 1 /k(h) 2 1 k (a) 2 /k(a) τ ii) III τ min H 0 (x) 1.1 iii) σ h (x) 0. 0 = Ω { A DΛ h H (µ r ) } h ρ h H s 1 H dx = Ω A (µ r ) h ρ h H s 1 H dx µ h + ρ h A r /H s 1 < 0 iv) σ a (x) 0 σ h (x) 0 v) 30 [21] ρ 0 = 0 (GM) t + (a(x, t), h(x, t)) (0, 0)

11 4. (A) 11 σ a (x) σ h (x) 0 σ a (1.1) (1.3) ( [17, 5, 8, 9, 10, 19, 12] ). σ a (x) x 1 Λ a = Λ h = d 2 /dx 2, µ a (x) µ h (x) 1 ρ a (A, H, x) ρ h (A, H, x) 1 B a = B h = d/dx D + (1.1) (1.3) (1.1) D D + 2 H/ x 2 0 H x. (1.1) Ω τ d l dt 0 H dx = l 0 H dx + l 0 Ar /H s dx + l 0 σ h(x) dx. H(x, t) ξ(t), ξ(t). σ h (x) 0, : (1.18) (1.19) (1.20) A t = ε2 2 A x 2 A + Ap ξ q + σ a(x) (0 < x < l, t > 0), l τ dξ dt = ξ + 1 lξ s A r dx (t > 0), 0 A A (0, t) = (l, t) = 0 (t > 0). x x w : w w + w p = 0, w > 0 (0 < y < + ), (1.21) w (0) = 0, lim w(y) = 0. y + w y + : sup 0<y< e y w(y) < +. Φ(y) (1.22) Φ Φ + pw p 1 Φ + pw p 1 = 0, Φ (0) = 0, lim Φ(y) = 0. y + (0 < y < + ), (, [pp , 10] ). σ a [17] Theorem 1 σ a (x)

12 12 (March 13, 2010) 1.8. max 0 x l σ a (x) > 0. 0 < r < 1 min 0 x l σ a (x) > 0. ε 0 ε (0, ε 0 ) (A 1,ε (x), ξ 1,ε ), (A 2,ε (x), ξ 2,ε ) ε 0 (1.23) (1.24) (1.25) (1.26) A 1,ε (x) = ξ q/(p 1) 1,ε ξ 1,ε = { ε ( 1 l A 2,ε (x) = ξ q/(p 1) 2,ε ξ 2,ε = { ( x ) } ( x ) w + o(1) + σ a (x) + σ a (0)Φ + o(1), ε ε (p 1)/[qr (p 1)(s+1)] w(z) r dz + o(1))}, 0 { ( ) } ( ) l x l x w + o(1) + σ a (x) + σ a (l)φ + o(1), ε ε { ( 1 (p 1)/[qr (p 1)(s+1)] ε w(z) r dz + o(1))}, l 0, (1.23) (1.25) o(1) x [0, l] (1.23) (1.25) σ a (x) Λ a = d 2 /dx 2, µ a (x) 1 (1.11) Ω = (0, l) Σ ε (x) : (1.27) ε 2 Σ ε Σ ε + σ a (x) = 0 (0 < x < l), Σ ε(0) = Σ a(l) = r = 2 1 < p < 5. α 0 α (0, α 0 ) ε 1 > 0 0 < ε < ε 1 τ 1 > 0 τ 2 > 0 (i) 0 < τ < τ 1 (A 1,ε (x), ξ 1,ε ) ; 0 < τ < τ 2 (A 2,ε (x), ξ 1,ε ) ; (ii) τ > τ 1 (A 1,ε (x), ξ 1,ε ). τ > τ 2 (A 2,ε (x), ξ 1,ε ).

13 (A) < p < 5 r = p + 1. α ε > 0 (p, q, s) ε 0 < τ 2,1 < τ 1,1 0 < τ 2,2 < τ 1,2 : (i) τ 2,1 < τ < τ 1,1 (A 1,ε (x), ξ 1,ε ) ; τ 2,2 < τ < τ 1,2 (A 2,ε (x), ξ 1,ε ) ; (ii) τ > τ 1,1 (A 1,ε (x), ξ 1,ε ) τ > τ 1,2 (A 2,ε (x), ξ 1,ε ) r = 2 r = p + 1 ( [11] ) r σ a (x) σ a (x) σ a ρ a (A, H, x) Ren [13]

14 14 (March 13, 2010) [1] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972), [2] H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), [3] A. J. Koch and H. Meinhardt, Biological pattern formation from basic mechanisms to complex structures, Rev. Modern Physics 66 (1994), [4] M. Li, S. Chen and Y. Qin, Boundedness and blow up for the general activator-inhibitor model, Acta Math. Appl. Sinica 11 (1995), [5] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), [6] K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math. 4 (1987), [7] W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations 299 (2006), [8] W.-M. Ni and I. Takagi, On the shape of least energy solution to a semilinear neumann problem, Comm. Pure Appl. Math. 44 (1991), [9] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear neumann problem, Duke Math. J. 70 (1993), [10] W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math. 12 (1995), [11] W.-M. Ni, I. Takagi and E. Yanagida, Stability analysis of point condensation solutions to a reaction-diffusion system, preprint. [12] W.-M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math. 18 (2001), [13] X. Ren, Least-energy solutions to a non-autonomous semilinear problem with small diffusion coefficient, Electron. J. Differential Equations 1993 (1993), [14] F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math. 1072, Springer, [15] K. Suzuki and I. Takagi, On the role of the source terms in an activator-inhibitor system proposed by Gierer and Meinhardt, Adv. Stud. Pure Math. 47-2, Math. Soc. Japan, Tokyo, 2007, [16] K. Suzuki and I. Takagi, On the role of basic production terms in an activator-inhibitor system modeling biological pattern formation, preprint, 2010.

15 (A) 15 [17] I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), [18] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London, Ser. B 237 (1952), [19] J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Differential Equations, 134 (1997), [20] J. Wu and Y. Li, Classical global solutions for the activator-inhibitor model, Acta Math. Appl. Sinica 13 (1990), [21] Niro Yanagihara, private communications.

第5章偏微分方程式の境界値問題

第5章偏微分方程式の境界値問題 October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ