American Institute of Mathematical Sciences

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August  2011, 30(3): 791-806. doi: 10.3934/dcds.2011.30.791

Global bifurcation and stable two-phase separation for a phase field model in a disk

Yasuhito Miyamoto 1,

1. 

Department of Mathematics, Tokyo Institute of Technology, O-Okayama, Meguro-ku, Tokyo 152-8551, Japan

Received  January 2010 Revised  October 2010 Published  March 2011

Let D:$=\{(x,y);\ x^2+y^2 < l^2\}\subset\R^2$. We study the shape of a local minimizer of the problem

$E(u)=\int_{D}(\frac{|\nabla u|^2}{2}+\lambda W(u)) dx\ \ $subject to$ \ \ m=\frac{1}{|D|}\int_Dudx $

and study the global structure of critical points. We show that for an arbitrary potential $W\in C^4$ every level set of every nonconstant local minimizer is a $C^1$-curve and it divides $D$ into exactly two simply connected subdomains. Next we consider the case $W(u)=(u^2-1)^2/4$ (Cahn-Hilliard equation). When $\lambda$ varies and $m$ is fixed, we show that this problem has an unbounded continuum of critical points. When $m$ varies and $\lambda$ is fixed, we show that this problem has a bounded continuum meeting at two different points on the trivial branch. Moreover, we show that in each case a bifurcating critical point is stable (a local minimizer) near the bifurcation point in a certain parameter range. The main technique is the nodal curve which relates the shape with the Morse index. We do not use a small parameter or the $\Gamma$-convergence technique.

Keywords: Phase separation; Cahn-Hilliard equation; Local minimizer; Second eigenvalue; Nodal curve..
Mathematics Subject Classification: Primary: 35B32, 35B36; Secondary: 35J20, 35J6.
Citation: Yasuhito Miyamoto. Global bifurcation and stable two-phase separation for a phase field model in a disk. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 791-806. doi: 10.3934/dcds.2011.30.791
References:
[1]

J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval,, Arch. Rational Mech. Anal., 86 (1984), 317. doi: 10.1007/BF00280031. Google Scholar

[2]

R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions,, J. Differential Equations, 27 (1978), 266. doi: 10.1016/0022-0396(78)90033-5. Google Scholar

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P. Freitas and C. Rocha, Lyapunov functionals and stability for FitzHugh-Nagumo systems,, J. Differential Equations, 169 (2001), 208. doi: 10.1006/jdeq.2000.3901. Google Scholar

[4]

M. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids,, Quart. Appl. Math., 46 (1988), 301. Google Scholar

[5]

M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments,, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 351. Google Scholar

[6]

B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains,, Comm. Math. Phys., 202 (1999), 629. doi: 10.1007/s002200050599. Google Scholar

[7]

P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations,, Amer. J. Math., 75 (1953), 449. doi: 10.2307/2372496. Google Scholar

[8]

H. Kielhöfer, Pattern formation of the stationary Cahn-Hilliard model,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1219. Google Scholar

[9]

H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs,", Applied Mathematical Sciences, 156 (2004). Google Scholar

[10]

S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: proof by the complete elliptic integrals,, Discrete Contin. Dyn. Syst., 19 (2007), 609. doi: 10.3934/dcds.2007.19.609. Google Scholar

[11]

S. Luckhaus and L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions,, Arch. Rational Mech. Anal., 107 (1989), 71. doi: 10.1007/BF00251427. Google Scholar

[12]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401. doi: 10.2977/prims/1195188180. Google Scholar

[13]

H. Matano, Private communication, (2005)., (2005). Google Scholar

[14]

B. McCartin, Eigenstructure of the equilateral triangle. II. The Neumann problem,, Math. Probl. Eng., 8 (2002), 517. doi: 10.1080/1024123021000053664. Google Scholar

[15]

Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball,, J. Differential Equations, 229 (2006), 494. doi: 10.1016/j.jde.2006.03.015. Google Scholar

[16]

Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball II,, J. Differential Equations, 239 (2007), 61. doi: oi:10.1016/j.jde.2007.05.006. Google Scholar

[17]

Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains,, Quart. Appl. Math., 65 (2007), 357. Google Scholar

[18]

Y. Miyamoto, Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk,, J. Funct. Anal., 256 (2009), 747. doi: 10.1016/j.jfa.2008.11.023. Google Scholar

[19]

Y. Miyamoto, The "hot spots" conjecture for a certain class of planar convex domains,, J. Math. Phys., 50 (2009). Google Scholar

[20]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rational Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230. Google Scholar

[21]

S. Maier-Paape and U. Miller, Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square,, Discrete Contin. Dyn. Syst., 15 (2006), 1137. doi: 10.3934/dcds.2006.15.1137. Google Scholar

[22]

Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit,, Dynamics Reported, 3 (1994), 25. Google Scholar

[23]

W. M. Ni, P. Poláčik and E. Yanagida, Monotonicity of stable solutions in shadow systems,, Trans. Amer. Math. Soc., 353 (2001), 5057. doi: 10.1090/S0002-9947-01-02880-X. Google Scholar

[24]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems,, Arch. Rational Mech. Anal., 101 (1988), 209. doi: 10.1007/BF00253122. Google Scholar

[25]

T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term,, Nonlinear Anal., 71 (2009), 1329. doi: 10.1016/j.na.2008.12.007. Google Scholar

[26]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081. Google Scholar

[27]

E. Yanagida, Private, communication (2006)., (2006). Google Scholar

show all references

References:
[1]

J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval,, Arch. Rational Mech. Anal., 86 (1984), 317. doi: 10.1007/BF00280031. Google Scholar

[2]

R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions,, J. Differential Equations, 27 (1978), 266. doi: 10.1016/0022-0396(78)90033-5. Google Scholar

[3]

P. Freitas and C. Rocha, Lyapunov functionals and stability for FitzHugh-Nagumo systems,, J. Differential Equations, 169 (2001), 208. doi: 10.1006/jdeq.2000.3901. Google Scholar

[4]

M. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids,, Quart. Appl. Math., 46 (1988), 301. Google Scholar

[5]

M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments,, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 351. Google Scholar

[6]

B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains,, Comm. Math. Phys., 202 (1999), 629. doi: 10.1007/s002200050599. Google Scholar

[7]

P. Hartman and A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations,, Amer. J. Math., 75 (1953), 449. doi: 10.2307/2372496. Google Scholar

[8]

H. Kielhöfer, Pattern formation of the stationary Cahn-Hilliard model,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1219. Google Scholar

[9]

H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs,", Applied Mathematical Sciences, 156 (2004). Google Scholar

[10]

S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: proof by the complete elliptic integrals,, Discrete Contin. Dyn. Syst., 19 (2007), 609. doi: 10.3934/dcds.2007.19.609. Google Scholar

[11]

S. Luckhaus and L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions,, Arch. Rational Mech. Anal., 107 (1989), 71. doi: 10.1007/BF00251427. Google Scholar

[12]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401. doi: 10.2977/prims/1195188180. Google Scholar

[13]

H. Matano, Private communication, (2005)., (2005). Google Scholar

[14]

B. McCartin, Eigenstructure of the equilateral triangle. II. The Neumann problem,, Math. Probl. Eng., 8 (2002), 517. doi: 10.1080/1024123021000053664. Google Scholar

[15]

Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball,, J. Differential Equations, 229 (2006), 494. doi: 10.1016/j.jde.2006.03.015. Google Scholar

[16]

Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball II,, J. Differential Equations, 239 (2007), 61. doi: oi:10.1016/j.jde.2007.05.006. Google Scholar

[17]

Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains,, Quart. Appl. Math., 65 (2007), 357. Google Scholar

[18]

Y. Miyamoto, Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk,, J. Funct. Anal., 256 (2009), 747. doi: 10.1016/j.jfa.2008.11.023. Google Scholar

[19]

Y. Miyamoto, The "hot spots" conjecture for a certain class of planar convex domains,, J. Math. Phys., 50 (2009). Google Scholar

[20]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rational Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230. Google Scholar

[21]

S. Maier-Paape and U. Miller, Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square,, Discrete Contin. Dyn. Syst., 15 (2006), 1137. doi: 10.3934/dcds.2006.15.1137. Google Scholar

[22]

Y. Nishiura, Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit,, Dynamics Reported, 3 (1994), 25. Google Scholar

[23]

W. M. Ni, P. Poláčik and E. Yanagida, Monotonicity of stable solutions in shadow systems,, Trans. Amer. Math. Soc., 353 (2001), 5057. doi: 10.1090/S0002-9947-01-02880-X. Google Scholar

[24]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems,, Arch. Rational Mech. Anal., 101 (1988), 209. doi: 10.1007/BF00253122. Google Scholar

[25]

T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term,, Nonlinear Anal., 71 (2009), 1329. doi: 10.1016/j.na.2008.12.007. Google Scholar

[26]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081. Google Scholar

[27]

E. Yanagida, Private, communication (2006)., (2006). Google Scholar

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