鈴木, 宏昌

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Title周期的な反応を考慮した相転移モデルの定常パターンについて (反応拡散系におけるパターン形成と漸近的幾何構造の研究)Author(s)鈴木, 宏昌Citation数理解析研究所講究録 (2004), 1356: 91-99Issue Date2004-02URL http://hdl.handle.net/2433/25187RightType Departmental Bulletin PaperTextversionpublisherKyoto University91(Hiromasa Suzuki)Faculty of Education,Shiga University1, , ,[5]. , Cahn-Hiffiard, .[5] , 2 traveling lamellar pattern, traveling hexagonalpattern , . ,, .1.1[5] , . $A,B,C$$Aarrow^{\gamma 1}B\gamma_{2}arrow C\gamma_{3}arrow A$, ( ) . $\psi_{A},$ $\psi_{B},$ $\psi c$$A,B,C$ , $\gamma_{i}>0$ $(\mathrm{i}=1,2,3)$ ,.$\frac{\partial\psi_{A}}{\partial t}=-\gamma_{1}\psi_{A}+\gamma_{3}\psi_{C}$$\frac{\partial\psi_{B}}{\partial t}=-\gamma_{1}\psi_{B}+\gamma_{3}\psi_{A}$$\frac{\partial\psi_{C}}{\partial t}=-\gamma_{1}\psi_{C}+\gamma_{3}\psi_{B}$$\psi_{A}+$ teB $+\psi_{C}=1$, $\psi=\psi_{A}-\psi_{B},$ $\phi=\psi_{A}+\psi_{B}$ , $\psi$ $\phi$$h_{1}$ $(\psi, \phi)$ $:=a_{11}\psi+a_{12}\phi$$h_{2}(\psi, 6)$ $:=a_{21}\psi+a_{22}\phi$1356 2004 91-9992.$(\begin{array}{ll}a_{\mathrm{l}1} a_{12}a_{21} a_{22}\end{array}).=(-(\gamma_{1}+-\mathrm{z}_{2}\mathrm{a}\mathrm{L}2)2$ $-(\gamma_{12}-\mathrm{L}2-(_{2}\lambda 2+\gamma \mathrm{a})+\gamma\epsilon))$.(1.1) $\{$$\frac{\partial\psi}{\partial t}=\nabla$ . $(M_{1} \nabla\frac{\delta F}{\delta\psi})+h_{1}(\psi, \phi)$$\frac{\partial\phi}{\partial t}=\nabla$ . $(M_{2} \nabla\frac{\delta F}{\delta\phi})+h_{2}(\psi, \phi)$$F= \int d\mathrm{r}[\frac{D_{1}}{2}|\nabla\psi|^{2}+\frac{D_{2}}{2}|\nabla\phi|^{2}+w(\psi, \phi)]$. [5] , $M_{1}=M_{2}=1,$ $D_{2}=0$ , $\phi$$w( \psi, \phi)=-\frac{\tau}{2}\psi^{2}+\frac{1}{4}\psi^{4}$$\{$$\frac{\partial\psi}{\partial t}=\nabla^{2}$ [-d$\nabla^{2}\psi-\tau\psi+\psi^{3}$] $+h1$ $(\psi, \phi)$ ,$\frac{\partial\phi}{\partial t}=h_{2}(\psi, \phi)$. $\tau$, [5] .1.2, 8(1.2)$\frac{\partial u}{\partial t}=\Delta$ ( $-\epsilon^{2}\Delta$u-f$(u)$ ) $+a_{11}u+a_{12}w+\gamma_{3}$ ,$x\in\Omega$ , $t>0$ ,$\frac{\partial w}{\partial t}=a_{2}1u+a22w$ $+\gamma$3,(1.3) $\frac{\partial u}{\partial n}=\frac{\partial(\Delta u)}{\partial n}=0$ , $\frac{\partial w}{\partial n}=0$ on $\partial\Omega$. $f(u)=u-u^{3}$ , $\epsilon$ , $\Omega\subset \mathrm{R}^{N}(N\geq 1)$ . $\epsilon$, (strong segregation limit).83, $v$ (x, $t$) (1.2)-(1.3)(1.4) $\{$$\frac{\partial u}{\partial t}=-\Delta v+a11u$ $+a12w$ $+\gamma$3,$0=\epsilon^{2}\Delta$u-f$(u)-v$ , $x\in\Omega$ , $t>0$ ,$\frac{\partial w}{\partial t}=a_{2}1u+a22w$$+\gamma$3(1.5) $\frac{\partial u}{\partial n}=0$ , $\frac{\partial v}{\partial n}=0$ , $\frac{\partial w}{\partial n}=0$ $x\in\partial\Omega$$\langle$$\Gamma$ (1.4)-(1.5) ,.. 1 (OJ) : .., 1 ,, 1 , a.1(1.6) $\{$$\epsilon^{2}$ux$x+f(u)-v=0$,$v_{xx}-a_{11}u-a_{12}w-\gamma_{3}=0,0<x<L$ ,$a_{21}u+a_{22}w+\gamma_{3}=0$u $(0)=0=u_{x}$ (L), $v_{x}(0)=0=v_{x}$ (L), $w_{x}(0)=0=w_{x}$ (L).. (1.6) 3 $w$ , 2(1.7)’$\epsilon$2u$xx+f(u)-v=0$,$v_{xx}+\alpha u+\beta=0$ ,,$\alpha=\frac{a_{12}a_{21}-a_{11}a_{22}}{a_{22}}>0,$ $\beta=(\frac{a_{12}}{a_{22}}-1)\gamma_{3}$. (1.7) activator-inhibitor .(1.7) ,$(1.8)_{-}$ $\{$$v_{xx}^{-}+\alpha h^{-}(v^{-})+\beta=0,0<x<x_{0}$ ,$v_{l}^{-}(0)$ $=0,$ $v^{-}(x0)$ $=v^{*}$ ,94$(1.8)_{+}$ $\{$$v_{xx}^{+}+\alpha h+(v^{+})+\beta=0$ , $x_{0}<x<L$ ,$v^{+}(x_{0})=v$”, $v_{x}^{+}(L)$ $=0$ ,(1.9) $v$;(xo) $=v_{x}^{+}$ (xo).. $u=h^{\pm}(v)$ $f(u)-v=0$$h^{-}(v)<h^{0}(v)<h^{+}(v)$ ( 1 ), $v^{*}(=0)$$J(v)= \int_{h^{-}(v)}^{h^{+}(v)}(f(u)-v)du$1: $f\Subset$), $\alpha,$ $\beta,$ $L$ ( PropO-sition 2.1 ) $(1.8)_{\pm}-(1.9)$ $v^{\pm}(x)$ , $V^{*}(x)$ .$V^{*}(x)=\{$$v^{-}(x)$ $x\in(0, x_{0})$$v^{+}(x)$ $x\in(x_{0}, L)$, .Theorem 1.1. (1.8)\pm -(1.9) $V$“(x) . ,$\epsilon_{0}>0$ : $\epsilon\in$ $(0, \epsilon 0)$ (1.6) ($u^{\epsilon}(x),$ $v$ \epsilon (x), $w^{\epsilon}(x)$ ), .(i) $\lim_{\epsilon\prec 0}v^{\epsilon}(x)=V^{*}(x)uni$ formly on $[0, L]$(ii) $0< \delta<\min${ $x_{0},$ $L$ -x0} $\delta$$\lim_{\epsilon\prec 0}u^{\epsilon}(x)=\{$$h^{-}(V^{*}(x))$ on $[0, x_{0}-\delta]$$h^{+}(V^{*}(x))$ on $[x_{0}+\delta, L]$ .85($u^{\epsilon}(x),$ $v$ \epsilon (x), $w^{\epsilon}(x)$ ) , .(1.10) $\{$$\lambda^{\epsilon}p=-qxx+a11p$ $+$ a12r,$0=\epsilon^{2}\ovalbox{\tt\small REJECT}+f’(u^{\epsilon}(x))p-q$ ,$\lambda^{\epsilon}r=a_{21}p+a_{22}r$ ,(1.11) $p_{x}(0)=0=p_{x}$ (L), $q_{x}(0)$ $=0.=q_{x}$ (L), $r_{l}(0)=0=r_{x}$ (L)., .Theorem 1.2. $a_{12}<0,$ $(a_{22}-\tau^{*})^{2}+4a_{12}a_{21}\neq 0$ . $\epsilon_{0}>0$, $\epsilon\in$ $(0, \epsilon 0)$ (1.10)-(1.11) $\lambda^{\epsilon}$ ., ( $u^{\epsilon}(x),$ $v$ \epsilon (x), $w^{\epsilon}(x)$ ) . $\tau^{*}$ ,$\{$$oq_{ex}+(\tau^{\epsilon}-a_{11})p=0$,$\epsilon^{2}p_{xx}+f’(u^{\epsilon}(x))p-q=0$ ,px(0)=0=p (L), $q_{x}(0)=0=q_{x}(L)$$\tau^{\epsilon}$$\lim_{\epsilonarrow 0}\tau^{\epsilon}=\tau^{*}$ .2 , 3 . 4.2(1.8)\pm -(1.9) ; .Proposition 2.1. $u^{*}=-\beta/\alpha$ .(i) $\underline{u}<u^{*}<\overline{u}$ , $L_{0}>0$ , $L\in(0, L0)$ $L$ $(1.8)\pm^{-(1.9)}$.$(\mathrm{i}\mathrm{i})-1<u^{*}\leq\underline{u}$ $\varpi\leq u^{*}<1$ , $L>0$ $(1.8)_{\pm}-(1.9)$., $\underline{u},$ $\overline{u}(\underline{u}<\overline{u})$ $f’(u)=0$ 2 ( 1 )Proof. $\alpha h^{\pm}(v)+\beta$ (ii) (i) ., .Proof of Theorem 1.1. , [1], [2].863(1.10) 3 $r$ , 1(3.1) $\{$$q_{xx}+(\tau^{\epsilon}-a_{11})p=0$ ,$\epsilon^{2}$p$xx+f/$ ( $u\epsilon$(x))p-q $=0$ ,(3.2) $p_{x}(0)=0=p_{x}(L)$ , qx(0)=0=q (L)$arrow.*$(3.3) $\tau^{\epsilon}=\lambda^{\epsilon}-\frac{a_{12}a_{21}}{\lambda^{\epsilon}-a_{22}}$. (3.1)-(3.2) , $\int_{I}p$ (x)$dx=0$ ,. (3.1)-(3.2)$\tau^{\mathrm{e}}$ , .Theorem 3.1. $\epsilon_{0}>0$ , $\epsilon\in$ $(0, \epsilon 0)$ :(i)(3.1)-(3.2) $\tau^{\epsilon}$(ii) $\lim_{\epsilonarrow 0}\tau^{\epsilon}=\tau^{*}$ , $\tau^{*}<0$ .$\tau^{\epsilon}$ $\epsilonarrow 0$ , Theorem 1.2 .Proof of Theorem 1.2. (3.3) $\lambda^{\epsilon}$(3.4) $(\lambda^{\epsilon})^{2}-(\tau+a_{22})\lambda^{\epsilon}+\tau a_{22}-a_{12}a_{21}=0$ .(3.4) $\lambda^{\epsilon}$ 2 , $D$$D=(a_{22}-\tau^{\epsilon})^{2}+4a_{12}a_{21}$. $a_{ij}$ (i, $j=1,2$) $\tau^{\epsilon}$ , $\epsilon>0$$D>0$ $\lambda_{1}^{\epsilon}+\lambda_{2}^{\epsilon}=a_{22}+\tau^{\epsilon}<0$, $\lambda_{1}^{\epsilon}\lambda_{2}^{\epsilon}=a_{22}\tau^{\epsilon}-a_{12}a_{21}>0$$D<0$ $2{\rm Re}\lambda^{\epsilon}=a_{22}+\tau‘<0,$ $\lambda^{\epsilon}\overline{\lambda^{\epsilon}}=|\lambda|^{2}=a_{22}\tau^{\epsilon}-a_{12}a_{21}>0$. , $\lambda_{i}^{\mathrm{e}}$ $(i=1,2)$ (3.4) , $\lambda^{\epsilon}$. .Remark 3.2. ( ) , $a_{\dot{\iota}j}(i,j=1,2)$$a_{12}$ . , $\gamma_{k}$ $(k=1,2,3)$, $a_{12}>0$ ,.87Proof of Theorem 3.1. ,., $\tau^{\epsilon}$ . (3.1) 1 0 $L$ $x$$( \tau^{\epsilon}-a_{11})\int_{0}^{L}p(x)dx=0$.$\tau^{\epsilon}-a_{11}=0$ , $\tau^{\epsilon}=a_{11}<0$ . , $\tau^{\epsilon}-a_{11}>0$ $\int_{0}^{L}p$(x) $dx=0$.Sturm-Liouville .$\{$$L^{\epsilon}\phi:=\epsilon^{2}\phi_{xx}+f’(u^{\epsilon}(x))\phi=\zeta\phi$ , $x\in I$ ,$\phi_{x}(0)=0=\phi_{x}$ (L),$L^{\epsilon}$$\{\zeta_{n}^{\epsilon}\}_{n\geq 0}$ , $\{\phi_{n}^{\epsilon}\}_{n\geq 0}$ -(3.1) 2 $p$(3.5) $p=(L^{\epsilon})^{-1}q= \frac{\langle q,\phi_{0}^{\epsilon}\rangle}{\zeta_{0}^{\epsilon}}\phi 0+$ $(L^{\epsilon})\uparrow(q)$ , $(L^{\epsilon})^{\mathrm{t}}:= \sum_{n\geq 1}\frac{\langle,\phi_{n}^{\epsilon}\rangle}{\zeta_{n}}‘\phi_{n}^{\epsilon}$ .[2] .Lemma 3.3.(i) $\zeta_{0}^{\epsilon}>0>-\delta>\zeta_{1}^{\epsilon}>\zeta_{2}^{\epsilon}>\cdots$ , ($\delta>0$ $\epsilon$ )(ii) $\lim_{\epsilon\downarrow 0}\frac{\zeta_{0}^{\epsilon}}{\epsilon}=\hat{\zeta}_{0}^{*}>0.$Lemma 3.4. $h\in L^{2}(I)\cap L^{\infty}(I)$$\lim_{\epsilon\downarrow 0}(L^{\epsilon})^{\mathrm{t}}(h)=\frac{h}{f_{u}^{*}}$ , strongly in $L^{2}$ -sense., $f_{u}^{*}:= \lim_{\epsilonarrow 0}f’(u^{\epsilon}(x))$ .Lemma 3.5.$\lim_{\epsilon\downarrow 0}\frac{1}{\sqrt{\epsilon}}\phi_{0}^{\mathrm{e}}=c^{\iota}\delta_{x_{0}}$ $H^{-1}(I)$ -sense $(c^{*}>0)$ .(3.5) 0 $L$ $x$$\langle$q, $\phi 8/\sim\vee 7$) $\int_{0}^{L}\frac{1}{\zeta_{0}^{\epsilon}/\epsilon}\frac{\phi_{0}^{\epsilon}}{\sqrt{\epsilon}}dx+\int_{0}^{L}(L^{\epsilon})^{\mathrm{t}}(q)dx=0$Lemma 3.3-Lemma3.5 , .$q^{*}:= \frac{1}{c}*\lim_{\epsilonarrow 0}\langle q,$ $\phi$a/4).88(3.5) (3.1) 1(3.6) $q_{xx}+(\tau^{\epsilon}-a11)$ $[ \frac{\langle q,\phi_{0}^{\epsilon}/\sqrt{\epsilon}\rangle}{\zeta_{0}^{\epsilon}/\epsilon}.\phi_{0}^{\epsilon}/!+(L’)\uparrow(q)]=0$.$\tau^{\epsilon}=o(1)(\epsilonarrow 0)$ , (3.6) $\epsilonarrow 0$ .(3.7)q $\frac{a_{11}}{f_{u}^{l}}q=0$ , $x\in(0,x_{0})\cup(x_{0}, L)$$q_{x}(0)=0=q_{x}(L),$ $q(x_{0})=1$ ,$(c^{*})^{2}a_{11}$$o \lim_{e-*oe0+0}q_{l}(x)-\lim_{xarrow x0-0}q_{x}(x)=$$\hat{\zeta}_{0}^{*}$$q(x)/q$” $q$ (x) , $q(x_{0})=1$ . (3.7), .$\tau^{\epsilon}=O(1)(\epsilonarrow 0)$ , (3.6) .(3.8)$q_{xx}+ \frac{\tau^{*}-a_{11}}{f_{u}^{*}}q=0$ , $x\in$ (O, $x_{0}$ ) $\cup(x_{0}, L)$$q_{x}(0)=0=q_{x}(L),$ $q(x_{0})=1$ ,$\lim_{xarrow x\mathrm{o}+0}q_{x}(x)-\lim_{xarrow x0-0}q_{x}(x)=-\frac{(c^{*})^{2}(\tau^{*}-a_{11})}{\zeta_{0}^{*}}$.[2] Lemma 2.4 , (3.8) Neumann Dirichlet$q(0)=0=q(L)$$(3.8)_{D}$ , $(q, \tau^{*})=(q^{D}, 0)$ . , $D$ Dirichlet, [2] Lemma 2.5 , $x=x_{0}$[q ] $:= \lim_{xarrow x\mathrm{o}+0}oq_{e}(x)-\lim_{xarrow x\mathrm{o}-0}q_{x}(x)$$[q_{x}^{D}]<[q_{x}]<0$ ,(3.9) $<- \frac{(c^{*})^{2}(\tau^{*}-a_{11})}{\zeta_{0}^{\mathrm{r}}}$. Lemma 3.5 $c^{*}$ Sturm-Liouville$\epsilon$$\hat{\zeta}_{0}^{*}$ ,, (3.9) $\tau^{*}<0$ . $\epsilon>0$ $\tau^{\epsilon}<0$ , [2].904, 1 , (1.4)-(1.5). ,. , Cahn-Hilliard(CH), (BC) ,, Cahn-Hilliard (1.1) $w($ \psi , $\phi)$$|\nabla u|$ $(D_{2}=0)$ . (CH) (BC),. , $a_{ij}$ ,Remark 32 .1 , (BC). , (BC) $k$.$=$ ([3])$|$ , ([4])[5] , (1.1) ., , strong segregation limit.[1] M.Mimura, M.Tabata and Y.Hosono, Multiple solutions of twxpoint boundary valueproblems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980),613-631.[2] Y.Nishiura and H.Fujii, Stability of singularly perturbed solutions to systems ofreaction diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726-1770.[3] Y.Nishiura and I.Ohnishi, Some aspect mathematical of the micrO-phase separationin diblock copolymers, Physica $\mathrm{D}84$ (1995), 31-39.[4] Y.Nishiura and H.Suzuki, Higher dimensional SLEP equation and applications tomorphological stability in polymer problems, Preprint (2003), submitted for publi-cation.[5] T. Okuzono and T. Ohta, Traveling waves in phffle-separating reactive mixtures,Phys. Rev. $\mathrm{E}67$ , 056211 (2003).

周期的な反応を考慮した相転移モデルの定常パターンについて (反応拡散系におけるパターン形成と漸近的幾何構造の研究)

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周期的な反応を考慮した相転移モデルの定常パターンについて (反応拡散系におけるパターン形成と漸近的幾何構造の研究)

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